Aero

NX Nastran Aeroelastic Analysis User’s Guide

Proprietary & Restricted Rights Notice

© 2007 UGS Corp. All Rights Reserved. This software and related documentation are proprietary to UGS Corp. NASTRAN is a registered trademark of the National Aeronautics and Space Administration. NX Nastran is an enhanced proprietary version developed and maintained by UGS Corp. MSC is a registered trademark of MSC.Software Corporation. MSC.Nastran and MSC.Patran are trademarks of MSC.Software Corporation. All other trademarks are the property of their respective owners.

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NX Nastran Aeroelastic Analysis User’s Guide

Contents

Fundamentals of Aeroelastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 Introduction to Aeroelastic Analysis and Design . . . Aerodynamic Data Input and Generation . . . . . . . . Aerodynamic Theories . . . . . . . . . . . . . . . . . . . . . . Interconnection of the Structure with Aerodynamics Static Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . Flutter Solution Techniques . . . . . . . . . . . . . . . . . . Dynamic Aeroelastic Analysis . . . . . . . . . . . . . . . . Aeroelastic Design Sensitivities and Optimization . .

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. 1-2 . 1-6 . 1-7 1-18 1-30 1-45 1-55 1-67

Aeroelastic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 Overview . . . . . . . . . . . . . . . . . . . . . . Aerodynamic Modeling . . . . . . . . . . . . Static Aeroelastic Analysis . . . . . . . . . . Flutter Analysis . . . . . . . . . . . . . . . . . Dynamic Aeroelastic Response Analysis

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. 2-2 . 2-2 2-21 2-23 2-26

Input Files for Aeroelastic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 Overview . . . . . . . . . . . . Executive Control Section Case Control Section . . . . Bulk Data Section . . . . . . Restarts . . . . . . . . . . . . .

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3-2 3-2 3-2 3-3 3-6

Output Features and Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 Overview . . . . . . . . . . . . . . . . . . . . Static Aeroelasticity . . . . . . . . . . . . Flutter . . . . . . . . . . . . . . . . . . . . . . Dynamic Aeroelasticity . . . . . . . . . . Design Sensitivity And Optimization

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4-2 4-2 4-4 4-6 4-8

Aeroelastic Solution Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 Overview . . . . . . . . . . . . . . . . . Solution Sequence Functions . . . Solution Sequence Structure . . . Aeroelastic Modules . . . . . . . . . Selected Aeroelastic Data Blocks

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. 5-2 . 5-2 . 5-4 5-20 5-22

Static Aeroelastic Analysis Sample Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FSW Airplane in Level Flight (Example HA144A) . . . . . . . . . . . . . Jet Transport Wing in Roll (Example HA144B) . . . . . . . . . . . . . . . A 15-Degree Sweptback Wing in a Wind Tunnel (Example HA144C)

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. 6-2 . 6-2 6-24 6-39

NX Nastran Aeroelastic Analysis User’s Guide

3

Contents

FSW Airplane in Antisymmetric Maneuvers (Example HA144D) . . . . . . . . . . . . . . . . FSW Airplane in Unsymmetric Quasi-Steady Maneuvers (Example HA144E) . . . . . . FSW Airplane with Bodies (Example HA144F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unit Solutions for Loadings of the FSW Airplane (Examples HA144GA and HA144GB)

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. 6-58 . 6-75 6-101 6-129

Flutter Analysis Sample Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 Three Degree of Freedom Airfoil and Fuselage (Example HA145A) . . . . . . . . . . . . . . . . . . 7-3 Flutter Analysis of Jet Transport Wing by Lifting Surface Theory (Example HA145B) . . . 7-16 Flutter and Divergence Analysis of Jet Transport Wing by Strip Theory (Example HA145C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-30 Flutter Analysis of Jet Transport Wing/Aileron by Strip Theory (Example HA145D) . . . . . 7-40 Subsonic Flutter Analysis of the 15-Degree Sweptback Wing by the KE-Method (Example HA145E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-50 Low Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using Mach Box Aerodynamics (Example HA145F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-62 Low Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using ZONA51 Aerodynamics (Examples HA145FA and HA145FB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-69 High Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using Piston Theory (Example HA145G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-83 Flutter Analysis of a Square Simply Supported Panel (Examples HA145HA and HA145HB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-99 Estimation of Dynamic Stability Derivatives (Example HA145I) . . . . . . . . . . . . . . . . . . 7-131 Servoelastic Stability Analysis of a Missile (Example HA110A) . . . . . . . . . . . . . . . . . . . 7-144 Aeroservoelastic Stability Analysis of a Missile (Example HA145J) . . . . . . . . . . . . . . . . 7-163 Aerothermoelastic Stability of a Wing (Examples HA153A and HA145KR) . . . . . . . . . . . 7-179 Aeroelastic Design Sensitivities and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2 Aeroelastic Optimization of FSW Airplane (Example HA200A) . . . . . . . . . . . . . . . . . . . . . 8-3 Aeroelastic Optimization of FSW Airplane (Example HA200B) . . . . . . . . . . . . . . . . . . . . 8-28 Dynamic Aeroelastic Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 Discrete Gust Response of BAH Wing (Example HA146A) . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 Transient Rolling of BAH Wing Due to Aileron (Example HA146B) . . . . . . . . . . . . . . . . . 9-16 Random Gust Response of BAH Wing (Example HA146C) . . . . . . . . . . . . . . . . . . . . . . . 9-25 Frequency Response of BAH Wing to Oscillating Aileron (Examples HA146D and HA146DR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-42 Subsonic Transient Response Analysis of a Sweptback Wing to an Impulsive Force Applied at the Tip (Example HA146E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-54 References and Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 Bibliography of Aeroelasticity and Unsteady Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . A-4 Bibliography of MSC.Nastran Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6 Commonly Used Commands for Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1 Executive Control Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Case Control Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 Aerodynamic Trim Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2

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NX Nastran Aeroelastic Analysis User’s Guide

Contents

Divergence Solution Selection . . . . . . . . . . . . . . . . Structural Damping and Transfer Function Selection Aeroelastic Flutter Solution Selection . . . . . . . . . . . Aeroelastic Dynamic Load Selection . . . . . . . . . . . . Output Control . . . . . . . . . . . . . . . . . . . . . . . . . . . Bulk Data Entries . . . . . . . . . . . . . . . . . . . . . . . . . Bulk Data Entry Summary . . . . . . . . . . . . . . . . . . Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B-2 B-2 B-2 B-3 B-3 B-3 B-3 B-5

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter

1 Fundamentals of Aeroelastic Analysis

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Introduction to Aeroelastic Analysis and Design

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Aerodynamic Data Input and Generation

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Aerodynamic Theories

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Interconnection of the Structure with Aerodynamics

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Static Aeroelasticity

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Flutter Solution Techniques

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Dynamic Aeroelastic Analysis

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Aeroelastic Design Sensitivities and Optimization

NX Nastran Aeroelastic Analysis User’s Guide

1-1

Chapter 1

1.1

Fundamentals of Aeroelastic Analysis

Introduction to Aeroelastic Analysis and Design

The NX Nastran Aeroelastic Analysis User’s Guide describes the theoretical aspects and the numerical techniques used to perform aeroelastic analyses with the software. You can use any of the existing NX Nastran structural finite elements (except axisymmetric elements) to build the structural model. NX Nastran generates the structural stiffness, mass, and damping matrices required by the aeroelastic analyses from your input of geometric, structural, inertial, and damping data, for subsequent use in the various aeroelastic analyses. The software computes matrices of aerodynamic influence coefficients from the data describing the geometry of the aerodynamic finite elements. The choice of aerodynamic grid points for the aerodynamic model is independent of the location of the structural grid points. One subsonic and three supersonic lifting surface aerodynamic theories are available in NX Nastran, as well as Strip Theory. The subsonic theory is the Doublet-Lattice method, which can account for interference among multiple lifting surfaces and bodies. The supersonic theories are the Mach Box method, Piston Theory, and the ZONA51 method for multiple interfering lifting surfaces. NX Nastran also provides an automated interpolation procedure to relate the aerodynamic to the structural degrees of freedom. Splining techniques for both lines and surfaces are used to generate the transformation matrix from structural grid point deflections to aerodynamic grid point deflections where local streamwise slopes are also computed. The transpose of this matrix transfers the aerodynamic forces and moments at aerodynamic boxes to structural grid points. The structural load distribution on an elastic vehicle in trimmed flight is determined by solving the equations for static equilibrium. The solution process leads to aerodynamic stability derivatives, e.g., lift and moment curve slopes and lift and moment coefficients due to control surface rotation, and trim variables, e.g., angle of attack and control surface setting, as well as aerodynamic and structural loads, structural deflections, and element stresses. The analysis at subsonic speeds utilizes the Vortex-Lattice aerodynamic theory (i.e., the steady case of the Doublet-Lattice method); the analysis at supersonic speeds uses the ZONA51 aerodynamic theory at zero reduced frequency. Control surface reversal speeds can be obtained by interpolation of roll control effectiveness, Clδ , versus flight dynamic pressure. Previously, static aeroelastic divergence speeds were determined by the K- or KE-methods of flutter analysis at very low reduced frequency or from the PK-method of flutter analysis. Now, those you can use the Divergence option in SOLutions 144 and 200 to obtain those speeds. The number of degrees of freedom required for accurate solutions to dynamic aeroelastic problems is generally far less than the number of physical degrees of freedom used in the finite element structural model. The number of independent degrees of freedom can be greatly reduced by using the (complex) amplitudes of a series of vibration modes as generalized coordinates, e.g., by Galerkin’s method. NX Nastran has the capability to compute the vibration modes and frequencies and to make the transformation to modal coordinates. The matrices of aerodynamic influence coefficients are also transformed to generalized aerodynamic forces by use of the vibration eigenvectors. The dynamic aeroelastic stability problem, flutter, is solved by any of three methods. The traditional American flutter method developed by the Air Materiel Command (AMC) in 1942 is available in the first two methods. •

The first method, called the K-method, is a variation of the AMC method.

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The second method, called the KE-method, is more efficient from the point of view of tracking roots, but is limited in input (no viscous damping) and output (no eigenvectors).

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NX Nastran Aeroelastic Analysis User’s Guide

Fundamentals of Aeroelastic Analysis

•

The third method, called the PK-method, is similar to the British flutter method, which was developed by the Royal Aircraft Establishment.

Historically, the capability to couple servo-systems with the structure has long been available in the software. With the addition of aerodynamic forces, aeroservoelastic analysis of stability augmentation or load alleviation systems is also available. The coupling with aerodynamic loads has also been added to the existing NX Nastran structural modal frequency response capability. Analyses of frequency response to arbitrarily specified forcing functions can be carried out using the oscillatory aerodynamic loads from any of the available aerodynamic theories. Frequency response to a harmonic gust field can be calculated at subsonic speeds using the Doublet-Lattice method for wing/body interference, and by the ZONA51 method for interfering lifting surfaces at supersonic speeds. Because unsteady aerodynamic loads are obtained only for steady-state harmonic motion, they are known only in the frequency- and not the time-domain. Inverse Fourier Transform techniques provide the appropriate methods by which transient response is obtained from the frequency response. Both forward and inverse Fourier transforms are provided so that the time-varying forcing function or the gust profile can be transformed into the frequency domain. Then, after convolution with the system frequency response, the inverse transform leads to the transient response of the system to the specified forcing function or gust profile. Stationary random response depends on the frequency response of the system to a specified loading and the power spectral density of that loading. The loading may be either a specified force distribution or a harmonic gust field. The statistical quantities of interest in the response ¯ the ratio of standard deviations (rms values) of the response to that of the input loading, are A, and N0, the mean frequency of zero crossings (with a positive slope) of the response. The capability to compute these quantities was added to NX Nastran by modifying the existing random response module to include options to generate various atmospheric turbulence power spectra and to perform the calculation of N0. The sensitivities of response parameters to changes in design variables are calculated by the perturbation techniques developed for structural optimization in NX Nastran also include static aeroelasticity and flutter. You can obtain basic aeroelastic sensitivities, including stability derivatives, trim variables, and flutter system dampings. The synthetic response technique of NX Nastran optimization also permits the calculation of sensitivities of user-specified functions of those standard response quantities. Optimization of aeroelastic characteristics can be combined with the other optimization features in SOLution 200, and vehicles can be designed optimally for aeroelastic loads, flying qualities, and flutter, as well as for strength, vibration frequencies, and buckling characteristics. In addition to the usual displacement and element force printout, you can also request certain output in graphical form. For checking the aeroelastic model, you can make of the layouts of structural and aerodynamic elements. You can also plot vibration mode shapes in one or more of three formats: •

grid line deflections

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grid point vector deflections

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contour lines.

You can make two-dimensional xy-type plots of flutter analysis results in the form of velocity-damping and velocity-frequency curves. You can plot frequency response data as functions of frequency and transient response data as functions of time. In random response

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 1

Fundamentals of Aeroelastic Analysis

analyses, you can plot the power spectral densities against frequency. NX Nastran offers several plotter options, the simplest being the printer that provides the results in a numerical format. A large number of problems is necessary to illustrate the principal features of the aeroelastic capability of NX Nastran. The analyses can be grouped under four headings: •

static aeroelasticity

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dynamic stability (flutter)

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dynamic response

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design sensitivity and optimization

Five static aeroelastic problems illustrate the symmetric, antisymmetric, and unsymmetric options. Flutter analyses by the three available flutter methods have then been variously selected to demonstrate all of the available aerodynamic theories. Examples of transient, frequency, and random responses are chosen as applications of the dynamic response analysis. Finally, several small examples are considered first for their design sensitivities and are then optimized for a variety of constraints on deflections, strength, and aeroelastic characteristics. A physical description of each problem, along with its finite element model, is given and a discussion of the results of each analysis is presented. Limited output is also presented for each example, including the input data echo and highlights of the calculated results. (The formats of the Executive Control, Case Control, and Bulk Data entries are presented in Input Files for Aeroelastic Problems (Ch. 4) and in Sections 3, 4, and 5 in the NX Nastran Quick Reference Guide.) The seven-character identification used in the Test Problem Library (TPL) is adopted for the example problems. The notation used here is HAXXXYZ where HA denotes Handbook for Aeroelastic Analysis [the title of the prior user document on aeroelasticity, see Rodden (1987)]. XXX denotes the Solution Sequence number (144, 145, 146 or 200), and Y is a letter denoting the specific example for a given Solution Sequence. Z, if used, denotes a specific feature of the example, e.g., R denotes a restart. To access the sample problems in the TPL, see: •

Using the Test Problem Libraries in the NX Nastran Installation and Operations Guide

Aeroelastic Modules and DMAP Sequences Aeroelastic analysis and design solution sequences extend the range of capabilities in NX Nastran beyond basic static and dynamic structural analysis. Modules are available to: •

Generate aerodynamic grid points

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Compute aerodynamic matrices

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Provide connection (interpolation) between the structural and aerodynamic grid points

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Solve the equations for static aeroelasticity

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Solve the equations for flutter

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Solve the equations for dynamic aeroelastic response

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Calculate aeroelastic design sensitivities

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Optimize aeroelastic and related structural characteristics

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NX Nastran Aeroelastic Analysis User’s Guide

Fundamentals of Aeroelastic Analysis

Four DMAP sequences are available: •

one for static aeroelastic analyses

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a second for modal flutter analyses by the K-, KE-, or PK-methods

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a third for modal dynamic aeroelastic response analyses due to gusts or control surface deflections

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a fourth for design sensitivity and optimization, including aeroelastic effects

Since this fourth sequence has applications to many areas other than aeroelasticity, this guide presents only design sensitivity and optimization information as it relates to aeroelasticity. For more comprehensive information, see: •

NX Nastran Design Sensitivity and Optimization User’s Guide

The Aerodynamic Finite Element Model NX Nastran aerodynamic analysis, like structural analysis, is based upon a finite element approach. The finite aerodynamic elements are strips or boxes on which there are aerodynamic forces. The aerodynamic elements, even for complex vehicles, must be in regular arrays. In particular, the aerodynamic elements for the lattice methods are arrays of trapezoidal boxes with sides that are parallel to the airflow. These can be described simply by defining properties of the array (panel). Because the grid points defining the structure usually do not coincide with the grid points defining the aerodynamic elements, provision has been made to generate equations for interpolating between the two. This interpolation is a key feature since it allows the choice of structural and aerodynamic elements to be based upon structural and aerodynamic considerations independently.

Oscillatory Aerodynamics Unsteady aerodynamic forces are generated when the flow is disturbed by the moving structure, or, as in the case of atmospheric turbulence, when the flow itself is unsteady. In the former case, theory leads to a matrix that relates the forces acting upon the structure due to the deflections of the structure. Methods that involve interactions among aerodynamic elements are available only for steady-state sinusoidal motion. Phase lags occur between the motions and the forces; thus, the elements of the matrices are complex numbers. Furthermore, these complex influence coefficient matrices depend upon two parameters of the flow: reduced frequency (dimensionless ratio of frequency to velocity) and Mach number (ratio of velocity to speed of sound). Such matrices, if computed by interaction theories such as the Doublet-Lattice or ZONA51 methods, are expensive to generate. An effective method to evaluate the matrices for a large number of parameter values is to compute the matrices for a few selected values and to interpolate to the remaining values. This parametric interpolation is an automatic feature of the solution modules for aeroelastic analysis. In the case of atmospheric turbulence theory leads to the forces on the structure in terms of the spectral composition of the turbulence, as well as the frequency response of the structure in a harmonic gust field. The turbulence spectrum and frequency response are used to generate the power spectra of selected responses and their statistical properties.

Aeroelastic Sensitivity and Optimization The sensitivities of aeroelastic responses require manipulating the equations used to perform the aeroelastic analysis while perturbing structural parameters, such as element areas and thicknesses. The result is a prediction of the change in a particular response, such as a stability

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 1

Fundamentals of Aeroelastic Analysis

derivative, due to a change in the parameter. This is useful in its own right to allow the user to gain insight into the aeroelastic design task and to make changes in the design in a systematic fashion. This capability is particularly useful in aeroelastic analyses since the aerodynamic effects interact with the structural stiffness and inertial properties in an often nonintuitive fashion. The sensitivities are most useful when coupled with the NX Nastran optimization capability. Here, you specify design goals (such as minimum weight) and restrictions (such as no flutter within the flight envelope), and the NX Nastran optimization procedure alters the design in order to meet these requirements. Because the aeroelastic optimization capability has been linked to the existing optimization capability, the design process can include requirements on nonaeroelastic factors such as a normal modal frequency or a stress response to a statically applied load that does not include aeroelastic effects.

1.2

Aerodynamic Data Input and Generation

Aerodynamic elements are strips, boxes, or segments of bodies that are combined to idealize the vehicle for the computation of aerodynamic forces. These elements, like structural elements, are defined by their geometry and their motions are defined by degrees of freedom at aerodynamic grid points. Requirements of the aerodynamic theory often dictate the geometry of the boxes. For example, the Mach Box method (MBM) uses only rectangular boxes, whereas the Doublet-Lattice (DLM) and ZONA51 methods assume trapezoidal boxes with their edges parallel to the free-stream velocity. By the use of aerodynamic input data, aerodynamic elements and grid points are automatically generated to help ensure that many of the theoretical requirements are met.

Aerodynamic Coordinate Systems Aerodynamic calculations are performed using a Cartesian coordinate system. By the usual convention, the flow is in the positive x-direction, and the x-axis of every aerodynamic element must be parallel to the flow in its undeformed position. (This is an assumption of aerodynamic small disturbance theory.) The structural coordinate systems may be defined independently, since the use of the same system for both may place an undesirable restriction upon the description of the structural model. You can specify any NX Nastran Cartesian system for the aerodynamic coordinates, as long as the flow is defined in the direction of the x-axis. All aerodynamic element and grid point data are transformed to the aerodynamic coordinate system. All the global (displacement) coordinate systems of the aerodynamic grid points will have their T1-directions in the flow direction. Their T3-directions will be normal to the element in the case of boxes, and parallel to the aerodynamic T2- and/or T3-directions in the case of bodies.

Aerodynamic Grid Points The aerodynamic grid points are physically located at the centers of the boxes for the lifting surface theories, at the centers of body elements for the DLM, at the quarter-chord/midspan point of the strips for Strip Theory and Piston Theory, and at user-defined points for the Mach Box method. Permanent constraints are generated for the unused degrees of freedom. A second set of grid points, used only for undeformed plotting, is located at the element corners. Grid point numbers are generated based upon the element identification number. For any panel, the box numbers start with the panel identification number and increase consecutively.

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Aerodynamic Degrees of Freedom Aerodynamic degrees of freedom, along with any extra points, are added after the structural matrices and modes have been determined. This introduces the following displacement sets for dynamic aeroelasticity:

uk usa up ups upa

Aerodynamic box and body degrees of freedom Permanently constrained degrees of freedom associated with aerodynamic grid points Physical degrees of freedom Union of up (physical) and usa Union of uk and ups (physical and aerodynamic)

The set upa replaces up as the set available for output at grid, scalar, and extra points. For static aeroelasticity, the sets are similar to the displacement sets in static analysis: ua uk

Structural analysis set Aerodynamic box and body degrees of freedom Aerodynamic “extra” points used to describe aerodynamic control surface deflections and overall rigid body motions, e.g., angle of attack or roll acceleration

ux

Finally, consider the j-set of aerodynamic control points. The j-set is not a user set; it is a notational set to identify aerodynamic matrices used in the solution processing. Physically, these are points on the structure where the downwash vectors are computed. As with the k-set, the location of these points is a function of the aerodynamic method employed: •

For Doublet-Lattice boxes, the control point is at the 75% chordwise station and spanwise center of the box

•

For ZONA51 boxes, the control point is at the 95% chordwise station and the spanwise center of the box

•

For Doublet-Lattice interference and slender body elements, the control point is along the axis of the element and at 50% of its length

•

For all other theories, the aerodynamic control points are at the same physical location as the aerodynamic grid points discussed above

1.3

Aerodynamic Theories

NX Nastran includes six aerodynamic theories: 1. Doublet-Lattice subsonic lifting surface theory (DLM)

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2. ZONA51 supersonic lifting surface theory 3. Subsonic wing-body interference theory (DLM with slender bodies) 4. Mach Box method 5. Strip Theory 6. Piston Theory Each of these methods is described in this section, but they all share a common matrix structure. Three matrix equations summarize the relationships required to define a set of aerodynamic influence coefficients [see Rodden and Revell (1962)]. These are the basic relationships between the lifting pressure and the dimensionless vertical or normal velocity induced by the inclination of the surface to the airstream; i.e., the downwash (or normalwash),

Equation 1-1. the substantial differentiation matrix of the deflections to obtain downwash,

Equation 1-2. and the integration of the pressure to obtain forces and moments,

Equation 1-3. where: wj

=

downwash

wj g

=

static aerodynamic downwash; it includes, primarily, the static incidence distribution that may arise from an initial angle of attack, camber, or twist

fj

=

pressure on lifting element j

=

flight dynamic pressure

k

=

reduced frequency, k = ωb/V where ω is the angular frequency, b is a reference semichord, and V is the free-stream velocity

Ajj (m, k)

=

aerodynamic influence coefficient matrix, a function of Mach number (m), and reduced frequency (k)

uk , P k

=

displacements and forces at aerodynamic grid points

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D1jk , D2jk

=

real and imaginary parts of substantial differentiation matrix, respectively (dimensionless)

Skj

=

integration matrix

The Aerodynamic Influence Coefficient Matrix The three matrices of Eq. 1-1, Eq. 1-2, and Eq. 1-3 can be combined to give an aerodynamic influence coefficient matrix:

Equation 1-4. All aerodynamic methods compute the S, D1, and D2 matrices at user-supplied Mach numbers and reduced frequencies. The Doublet-Lattice and ZONA51 theories compute the A matrix. Then, matrix decomposition and forward and backward substitution are used in the computation of the Q matrix. The remaining methods compute A−1 directly and use matrix multiplications to form Q.

Doublet-Lattice Subsonic Lifting Surface Theory The Doublet-Lattice method (DLM) can be used for interfering lifting surfaces in subsonic flow. The theory is presented by Albano and Rodden (1969), Giesing, Kalman, and Rodden (1971), and Rodden, Giesing, and Kalman (1972) and is not reproduced here. The following general remarks summarize the essential features of the method. The theoretical basis of the DLM is linearized aerodynamic potential theory. The undisturbed flow is uniform and is either steady or varying (gusting) harmonically. All lifting surfaces are assumed to lie nearly parallel to the flow. The DLM is an extension of the steady Vortex-Lattice method to unsteady flow. Each of the interfering surfaces (or panels) is divided into small trapezoidal lifting elements (“boxes”) such that the boxes are arranged in strips parallel to the free stream with surface edges, fold lines, and hinge lines lying on box boundaries. The unknown lifting pressures are assumed to be concentrated uniformly across the one-quarter chord line of each box. There is one control point per box, centered spanwise on the three-quarter chord line of the box, and the surface normalwash boundary condition is satisfied at each of these points. The code for computing the aerodynamic influence coefficients Ajj was taken from Giesing, Kalman, and Rodden (1972b). Any number of arbitrarily shaped interfering surfaces can be analyzed, provided that each is idealized as one or more trapezoidal planes. Aerodynamic symmetry options are available for motions which are symmetric or antisymmetric with respect to one or two orthogonal planes. The user may supply one-half (or one-fourth) of the model and impose the appropriate structural boundary conditions. The full aircraft can also be modeled when the aircraft or its prescribed maneuvers lack symmetry.

ZONA51 Supersonic Lifting Surface Theory ZONA51 is a supersonic lifting surface theory that accounts for the interference among multiple lifting surfaces. It is an optional feature in NX Nastran (available as the Aero II option). It is similar to the Doublet-Lattice method (DLM) in that both are acceleration potential methods that need not account for flow characteristics in any wake. An outline of the development of the

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acceleration-potential approach for ZONA51 is presented by Liu, James, Chen, and Pototsky (1991), and its outgrowth from the harmonic gradient method (HGM) of Chen and Liu (1985) is described. ZONA51 is a linearized aerodynamic small disturbance theory that assumes all interfering lifting surfaces lie nearly parallel to the flow, which is uniform and either steady or gusting harmonically. As in the DLM, the linearized supersonic theory does not account for any thickness effects of the lifting surfaces. Also, as in the DLM, each of the interfering surfaces (or panels) is divided into small trapezoidal lifting elements (“boxes”) such that the boxes are arranged in strips parallel to the free stream with surface edges, fold lines, and hinge lines lying on box boundaries. The unknown lifting pressures are assumed to be uniform on each box. There is one control point per box, centered spanwise on the 95 percent chord line of the box, and the surface normalwash boundary condition is satisfied at each of these points. The code for computing the aerodynamic influence coefficients, Ajj , was integrated into the software by Zona Technology, Inc., taking full advantage of the extensive similarities with the DLM. Any number of arbitrarily shaped interfering surfaces can be analyzed, provided that each is idealized as one or more trapezoidal planes. Aerodynamic symmetry options are available for motions that are symmetric or antisymmetric with respect to the vehicle centerline. Unlike the DLM, symmetry about the xy-plane is not supported. The user may supply one half of the vehicle model and impose the appropriate structural boundary conditions.

Subsonic Wing-Body Interference Theory The method of images, along with Slender Body Theory, has been added to the Doublet-Lattice method (DLM) in Giesing, Kalman, and Rodden (1972a, 1972b, and 1972c). The DLM is used to represent the configuration of interfering lifting surfaces, while Slender Body Theory is used to represent the lifting characteristics of each body (i.e., fuselage, nacelle, or external store). The primary wing-body interference is approximated by a system of images of the DLM trailing vortices and doublets within a cylindrical interference body that circumscribes each slender body. The secondary wing-body interference that results from the DLM bound vortices and doublets is accounted for by a line of doublets located on the longitudinal axis of each slender body. The boundary conditions of no flow through the lifting surfaces or through the body (on the average about the periphery) lead to the equations for the lifting pressures on the surfaces and for the longitudinal (and/or lateral) loading on the bodies in terms of the normalwashes on the wing-body combination. The code for computing the aerodynamic matrices was adapted for NX Nastran from Giesing, Kalman, and Rodden (1972b). The adaptation required a matrix formulation of all of the body interference and body loading calculations. These equations are written using the symbols adopted for NX Nastran and showing the equivalences to names used in the documentation of Giesing, Kalman, and Rodden (1972b). The program of Giesing, Kalman, and Rodden (1972b) finds the forces on the lifting boxes and bodies of an idealized airplane in terms of the motions of these elements. The lifting surfaces are divided into boxes. The bodies are divided into elements. There are two types of body elements: slender elements, which are used to simulate a body’s own motion, and interference elements, which are used to simulate the interaction with other bodies and boxes. The body elements may have z (vertical), y (lateral), or both degrees of freedom. The basic method is the superposition of singularities and their images. There are two basic singularity types: “forces” and modified acceleration potential “doublets.” Each “force” singularity is equivalent to a line of doublets in the wake. As discussed, the wing boxes use the “force” type of singularity concentrated along the box quarter chord. The interference elements use the “doublet” type of singularity. The slender body elements use both types. Downwashes are related to the singularities by

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Equation 1-5. where: ww

=

wing box downwashes at the three-quarter chord

ws

=

downwashes for slender body elements

fw

=

pressures concentrated along wing box quarter chords

μI

=

acceleration potential interference doublets

μs

=

acceleration potential slender element doublets (for flow fields only)

=

[DT] in Giesing, Kalman, and Rodden (1972b, §5.3.1)

= in Giesing, Kalman, and Rodden (1972b, §5.3.1) Ass

=

D2D−1, which is a diagonal matrix discussed below where

where AR = body cross section height-to-width ratio Eq. 1-6 relates the forces to the singularities:

Equation 1-6. where: Pw Ps Cs

= = =

wing box force body element force slender element forces per unit length divided by dynamic pressure

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Chapter 1

[Sww ] [Ssw SsI Sss ]

= =

Sss

=

box areas (a diagonal matrix) [BFS] in Giesing, Kalman, and Rodden (1972b, §5.8.1), but rearranged in the order of the rows Δx = body element length

Eq. 1-5 and Eq. 1-6 use Method 1 of Giesing, Kalman, and Rodden (1972b). All of the above matrices have been modified to include the images of the sources caused by the symmetry plane. In the slender body part of the program developed by Giesing, Kalman, and Rodden (1972b), there is no matrix that relates the slender element forces, Cs , to the slender element doublets, μs . This matrix had to be derived. This relationship between the forces and doublets involves only elements of the same body with the same orientation. The differential equation relating these distributions is

Equation 1-7. where: Cs (x) μs (x)

= =

x ω/U

= =

lift per unit length/dynamic pressure (which has units of length) velocity potential doublet strength per unit length/free stream velocity (which has units of length2) streamwise coordinate unit reduced frequency (which has units of length-1)

The elements of the vector {Cs } are C(xcenter )Δx, which are the total forces on the elements divided by the dynamic pressure. The elements of {μs } are μ(xcenter ). The values of Cs [called ΔCp ΔA by Giesing, Kalman, and Rodden (1972b)] are evaluated from an equation that is equivalent to

Equation 1-8. and

Equation 1-9. where: (SB)

=

ws a0 (x)

= =

1-12

downwash (dimensionless) half width (dimensional radius)

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AR

=

height/width ratio of body

The method used by Giesing, Kalman, and Rodden (1972b) leads to a matrix, [G], which relates the vector of {Cs } to μs

Equation 1-10. The derivation of [G] assumes that μs/a2o (x), which is proportional to ws , is a function of x. Thus, μs (x) = a2o (x)[μs (x)/a2o (x)]. Using Eq. 1-7,

Equation 1-11. The numerical derivatives required for the last term in Eq. 1-11 are evaluated by the following rules: One point per body: derivative = 0 Two or more points per body: The two point rule comes from a linear fit and the three point rule from a quadratic fit. Examples are:

Equation 1-12.

Equation 1-13. Using this, the elements of [G] for one body are given by

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Equation 1-14.

where is the reference chord for the reduced frequency k, and gij is tridiagonal:

Equation 1-15. For the first element (i = 1), the terms involving xx − 1 are deleted; for the last point (i = N), the terms involving xi + 1 are deleted. Eq. 1-10 is used to eliminate the slender element forces Cs from Eq. 1-6, giving

Equation 1-16. Eq. 1-5 and Eq. 1-16 permit relating the forces to the downwashes. The relationship of deflections to downwashes is given by Eq. 1-2. As can be seen from Eq. 1-5, there is zero “downwash” for all interference body elements, therefore, the rows of Djk 1 and Djk 2 associated with interference body elements vanish. All other rows represent total derivatives for downwashes of boxes and slender body elements. The basic form of Eq. 1-1 through Eq. 1-3 is kept, even in the case of panels with interference and slender bodies.

Mach Box Method The Mach Box method (MBM) is used to estimate generalized aerodynamic forces on an isolated planar wing that has up to two (adjacent) trailing edge control surfaces, a crank on the leading and trailing edges, and is oscillating in supersonic flow. The MBM is a modification of the square box method first proposed by Pines, Dugundji, and Neuringer (1955), outlined by Zartarian and

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Hsu (1955) and Zartarian (1956), and programmed by Moore and Andrew (1965) and by Donato and Huhn (1968). Donato and Huhn (1968) also present extensive comparisons of MBM results with exact theories. The general features of the method are summarized in the following remarks. The MBM is a numerical solution of the linearized three-dimensional oscillatory supersonic perturbation potential flow equation. The regions disturbed by the vibrating lifting surface are divided into a grid of rectangular lifting elements, called Mach boxes (i.e., rectangles with diagonals that are parallel to the Mach lines). The regions that are divided into Mach boxes include the wing and its control surfaces, as well as regions adjacent to the lifting surface that are within the envelope of aft Mach cones of the foremost points on the wing. Calculations are made of the influence of unit sources distributed uniformly over the area of each box on the velocity potential at the center of every box within its aft Mach cone. These are called velocity potential influence coefficients, and are functions of Mach number, reduced frequency, and relative locations of box centers. They are multiplied by the source strengths, defined by the normal velocity and perturbation angle of attack at the box center, and then summed to obtain the corresponding velocity potential at a downstream receiving point on the planform. When these velocity potential distributions are determined, they are multiplied by the complex conjugate of the source strength at the receiving point and summed over all such points to complete the calculation of the generalized aerodynamic force coefficients. This is the variation on A−1 discussed in Aerodynamic Theories ; the matrix Ajj in Eq. 1-1 is never computed in MBM. Instead of relating pressures to downwashes, the MBM relates velocity potentials to downwashes and thereby circumvents the errors associated with numerical differentiation of the potentials to obtain pressures. Then the generalized aerodynamic forces are obtained directly from the potentials (by using an integration by parts) without the intermediate step of finding the pressures. As with all potential theory methods, the accuracy of the MBM depends on the validity of supersonic linearized theory (which is generally assumed to be valid in the Mach number range from about 1.2 up to 3.0) and the number of Mach boxes with centers on the moving surface. At high Mach numbers the results approach those from first order Piston Theory.The supersonic Mach Box code used by NX Nastran is based on subroutines in a modified version of the program of Donato and Huhn (1968); the modifications were made by L. V. Andrew, G. V. Owens, and J. W. Sleison to include the second control surface.

Figure 1-1. Platform Geometry for Mach Box Method The general planform that can be analyzed is shown in Figure 1-1. The following options are available: •

Leading- and/or trailing-edge cranks

•

None, one, or two adjacent trailing-edge control surfaces with swaybacked hinge lines

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Fundamentals of Aeroelastic Analysis

Symmetric or antisymmetric motion

The free-stream velocity is parallel to the x-axis as shown in Figure 1-1 and a plane of symmetry is imposed at the inboard edge. The Mach Box computer algorithm searches for the extreme streamwise point and the extreme spanwise point on the wing/control surface planform. They may be different points. It then reduces the user-supplied integer number of chordwise boxes to the closest floating point value (v 50) that places the side edge of a Mach box on the extreme spanwise point. If this array satisfies the criterion of less than a total number of 500 boxes (with v 50 spanwise) that contribute to the integral of generalized forces (i.e., those partially on the main surface plus those cut by the trailing edge), and less than 200 boxes on each control surface that contribute to the integral, the grid is used to complete the analysis. If not, the number of chordwise boxes is sequentially reduced by one until both the edge criterion and the number criteria are satisfied, and then it proceeds to complete the analysis.

Strip Theory The first solutions to the unsteady theoretical aerodynamic problem were obtained in two dimensions by Theodorsen (1935) and by Küssner and Schwarz (1940). These solutions were utilized in flutter analyses by assuming that the loads at each spanwise station of a wing depended only on the motion of that station. An early method of flutter analysis, presented by Smilg and Wasserman (1942), divided the wing into a number of strips, and the aerodynamic loads on each strip were calculated on the basis of the two-dimensional coefficients evaluated at the centerline of the strip. This “Strip Theory” was surprisingly accurate in many cases, although its successes were primarily in applications to unswept wings with high aspect ratios and may have been the result of compensating errors in the application of two-dimensional theory to three-dimensional flow. Because of its place in tradition, Strip Theory is included in NX Nastran for use at your discretion. The NX Nastran code for computing Ajj −1 is based on Küssner and Schwarz (1940) and is an extension of a program written by E. Albano at the Northrop Corporation. Although Küssner and Schwarz (1940) include a trim tab, only the airfoil and an aerodynamically balanced control surface are considered in NX Nastran. Both the airfoil and the control surface are assumed to be rigid in the chordwise direction, and the control surface hinge line is assumed to remain on the wing chord line (i.e., no camber motions or hinge failures are considered). The Theodorsen function for the unsteady circulatory loads is written as

Equation 1-17. where k = ωb/V is the local reduced frequency and b is the semichord of the strip. The user has the option of invoking F(k) and G(k) from their exact expressions in terms of Bessel functions, or from approximations of the form

Equation 1-18.

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Equation 1-19. in which β0 = 0. The choice of values for the parameters bn and βn is left to the user. In this way, Strip Theory can be adjusted to account for compressibility or aspect ratio effects. Some values of bn and βn are tabulated on Bisplinghoff, Ashley, and Halfman (1955, pp. 350, 394). An approximate sweep correction is also incorporated; the correction is the factor cos , where is the one-quarter chord line sweep angle for the aerodynamic macro-element defined on the CAERO4 Bulk Data entry and is applied as a multiplier to all loads acting on the element.

Piston Theory In the limit of high Mach number (m2 » 1) or high reduced frequency (m2k2 » 1), the three-dimensional, pressure-downwash relationship on a lifting surface becomes a nonlinear uncoupled relationship at each point. The nonlinear point relationship can be linearized for small disturbances while retaining the nonlinear aspects of the initial steady-state condition. The result is known as third-order Piston Theory and was developed by Ashley and Zartarian (1956). A computer program to obtain Ajj −1 was written by Rodden, Farkas, Malcom, and Kliszewski (1962). The program uses much of the same logic as Strip Theory discussed previously. This computer code has been added directly to NX Nastran with only minor adjustments for sign conventions and has the following general features: it is an extension of Ashley and Zartarian (1956) to account for sweep and steady angle of attack and to decrease the lower supersonic Mach number limit so that agreement with Van Dyke (1952) is obtained through the second-order terms. A rigid chord is assumed as well as a rigid trailing edge control surface hinged at its leading edge (i.e., no aerodynamic balance is considered because that is not a design feature on supersonic vehicles). Experimental correlations have indicated the validity of Piston Theory in the range of Mach numbers from about 2.5 to 7.0.

Experimental Aerodynamic Corrections The theoretical aerodynamic pressures are found from Eq. 1-1.

Equation 1-20. Two experimental correction may be introduced into Eq. 1-3, so that the corrected force distribution becomes

Equation 1-21. where:

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a matrix of empirical correction factors to adjust each theoretical aerodynamic box [Wkk ] = lift and moment to agree with experimental data for incidence changes; Giesing, Kalman, and Rodden (1976) suggest one way of obtaining these factors. =

vector of experimental pressure coefficients at some reference incidence (e.g., zero angle of attack) for each aerodynamic element.

Both corrections are appropriate in static aeroelastic analysis and provision for them has been made and is illustrated in a sample problem for static aeroelasticity in Static Aeroelastic Analysis Sample Problems. However, the additive correction is not appropriate in the dynamic aeroelastic analyses since these are perturbation analyses. The multiplicative correction is illustrated in a sample problem for flutter analysis in Fundamentals of Aeroelastic Analysis. Correction terms are input using DMI Bulk Data entries with names corresponding to those given in the sample problems.

1.4

Interconnection of the Structure with Aerodynamics

Structural and aerodynamic grids are connected by interpolation. This allows the independent selection of grid points of the structure and aerodynamic elements of the lifting surfaces/bodies in a manner best suited to the particular theory. The structural model for a wing may involve a one-, two- or three-dimensional array of grid points. The aerodynamic theory may be a lifting surface theory or a strip theory. A general interpolation method is available that will interconnect the various combinations. Any aerodynamic panel or body can be subdivided into subregions for interpolation, using a separate function for each. The interpolation method is called splining. The theory involves the mathematical analysis of beams and plates (see Figure 1-1). Three methods are available: •

linear splines, which are a generalization of an infinite beam and allow torsional as well as bending degrees of freedom

•

surface splines, which are solutions for infinite uniform plates

•

an explicit user-defined interpolation

Several splines, including combinations of the three types, can be used in one model. For example, a model may use one spline for the horizontal tail and three splines for the wing (a surface spline for the inboard section, and a linear spline for the outboard wing section, and the explicit interpolation for the aileron). Separation into subregions allows discontinuous slopes (e.g., at the wing-aileron hinge), separate functions (for wing and tail), and smaller regions. Smaller regions reduce the computing time and may increase the accuracy. The structural degrees of freedom have been chosen in NX Nastran as the independent degrees of freedom; the aerodynamic degrees of freedom are dependent. A matrix is derived that relates the dependent degrees of freedom to the independent ones. The structural degrees of freedom may include any grid components. Two transformations are required: the interpolation from the structural deflections to the aerodynamic deflections and the relationship between the aerodynamic forces and the structurally equivalent forces acting on the structural grid points. The splining methods lead to an interpolation matrix [Gkg ] that relates the components of structural grid point deflections {ug } to the deflections of the aerodynamic grid points {uk },

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Equation 1-22. The derivation of the elements of [Gkg ] is discussed in the following sections for the surface spline and linear spline interpolation methods. Attention is first given to the transformation between the aerodynamic and structural force systems. This transformation is found from the requirement that the two force systems be “structurally equivalent” rather than statically equivalent. Structural equivalence means that the two force systems deflect the structure equally. Statically equivalent force systems, as used on a whiffletree in a static structural strength test, do not result in equal deflections. It is the deflections rather than resultant loads that are of primary interest in aeroelasticity. The concept of structural equivalence is discussed by Schmitt (1956) and Rodden (1959b).

Figure 1-2. Splines and Their Coordinate Systems

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The aerodynamic forces {Fk } and their structurally equivalent values {Fg } acting on the structural grid points therefore do the same virtual work in their respective deflection modes,

Equation 1-23. where δuk and δug are virtual deflections. Substituting Eq. 1-22 into the left-hand side of Eq. 1-23 and rearranging yields

Equation 1-24. from which the required force transformation is obtained because of the arbitrariness of the virtual deflections.

Equation 1-25. Equations Eq. 1-22 and Eq. 1-25 are both required to complete the formulation of aeroelastic problems in which the aerodynamic and structural grids do not coincide, i.e., to interconnect the aerodynamic and structural grid points. However, since the transpose of the deflection interpolation matrix is all that is required to connect the aerodynamic forces to the structure, it is appropriate in further discussions to refer to the interconnection problem as simply the problem of interpolating from the structural to the aerodynamic grid points.

Theory of Surface Splines A surface spline is a mathematical tool used to find a surface function, w(x, y), for all points (x, y) when w is known for a discrete set of points, wi = w(xi , yi ). The theory introduces an infinite plate and solves for its deflections, given its deflections at a discrete set of points; i.e., it is the problem of a plate with multiple deflecting supports. The surface spline is a smooth continuous function that will become nearly linear in x and y at large distances from the points (xi , yi ). This problem can be solved in closed form. The deflection of the plate is synthesized as the sum of deflections due to a set of point loads on the infinite plate. The deflection due to a single concentrated load is the fundamental solution and has polar symmetry. If the load is taken at xi = yi = 0, and polar coordinates are used (x = r cosθ, y = r sinθ), the governing differential equation is

Equation 1-26.

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The distributed load q vanishes except near r = 0. The general solution to the homogeneous form of Eq. 1-26 is

Equation 1-27. Set C2 = 0 to keep the solution finite as r → 0. Then multiply Eq. 1-26 by 2πr and integrate from r = 0 to r = ε (a small number) to obtain the concentrated force P,

Equation 1-28. Combining Eq. 1-27 and Eq. 1-28 leads to C3 = P/(8πD). The fundamental solution may therefore be written

Equation 1-29. since

The fundamental solutions are superimposed to solve the entire plate problem with a solution of the form

Equation 1-30. where ri 2 = (x − xi )2 + (y − yi )2. The remaining requirement is the satisfaction of the boundary condition at infinity: Radial lines emanating from loaded points (which all may be regarded as at the origin relative to infinity) appear to be straight lines. To do this, Eq. 1-30 is expanded in a series, assuming a large argument (x2 + y2), and delete all terms of order (x2 + y2)ln(x2 + y2), (x2 + y2), x ln (x2 + y2), and y ln (x2 + y2), leaving terms of order x,y, ln(x2 + y2), and 1. The details of the series expansion are given by Harder, MacNeal, and Rodden (1971, App. E). The deletion of the higher-order terms is accomplished by requiring

Equation 1-31.

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Equation 1-32.

Equation 1-33.

Equation 1-34. Eq. 1-31 through Eq. 1-34 result in linear deflections at infinity; Eq. 1-32 through Eq. 1-34 are also recognized as the equations of equilibrium. From Eq. 1-31 it is seen that

Equation 1-35. A solution to the general spline problem, formed by superimposing solutions of Eq. 1-26, is given by

Equation 1-36. where: Ki (x, y) ri 2 Pi

= = =

(1/(16πD))ri 2 ln ri 2 (x − xi )2 + (y − yi )2 concentrated load at (xi , yi )

The N + 3 unknowns (a0 , a1 , a2 , Pi ; i = 1, N) are determined from the N + 3 equations

Equation 1-37. and

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Equation 1-38. where Kij = Ki (xj , yj ). Note that Kij = Kji , and Kij = 0 when i = j. The above derivation is also summarized by Harder and Desmarais (1972a) and an application is shown. It is discussed further by Rodden, McGrew, and Kalman (1972) and by Harder and Desmarais (1972b). Eq. 1-36 can be rewritten in the matrix form:

Equation 1-39. Combining Eq. 1-36 and Eq. 1-37 into the matrix form

Equation 1-40. permits solution for the vector of ai and Pi . The interpolation to any point in the (x, y) plane is then achieved by evaluating w(x, y) from Eq. 1-39 at the desired points. This gives an overall equation of the form

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Equation 1-41. Slopes of the aerodynamic panels, which are the negative of the slopes of the displacements, are also required. These are found by analytically differentiating Eq. 1-41 with respect to x,

Equation 1-42. where:

Equation 1-43. Theory of Linear Splines A linear spline is a “beam” function, w(x), which passes through the known deflections, wi = w(xi ) with twists φi = φ(xi ). Bending deflections are easily solved by the three-moment method, which is appropriate for simple beams. However, an extension of the method is required for splines with torsion, rigid arms, and attachment springs. The following derivations outlined are based on analogy with the surface spline derivation. Bending Bars Equation for deflection

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Equation 1-44. where q is a distributed transverse load and M is a distributed moment. A symmetric fundamental solution for x ≠ 0 is used for lateral loads q = Pδ(x), where δ(x) is the Dirac δ-function, and an antisymmetric fundamental solution is used for moments. The solution for the general case is found by superimposing the fundamental solutions,

Equation 1-45.

Equation 1-46. These are written in matrix notation as

Equation 1-47. To satisfy the boundary condition at infinity, w(x) must approach a linear function. This requires

Equation 1-48.

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Equation 1-49. These are recognized as the equations of equilibrium. The unknowns ai , Pi , and Mi are found from

Equation 1-50. where it has been assumed that x1 < x2 ... <xN , and

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Torsion Bars Equation for twist:

Equation 1-51. where t is a distributed torque. The solution is

Equation 1-52. To satisfy the condition that φ = constant for large x requires the equilibrium condition

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Fundamentals of Aeroelastic Analysis

Equation 1-53. Then the unknowns a0 and Ti are found by solving

Equation 1-54.

Attachment of Splines with Elastic Springs If a number of grid points are located closely, forcing the splines through each deflected point may cause some erratic behavior at a distance from the cluster. For this reason elastic spring connections to smooth the spline have been introduced. The change in the formulas for splines to accommodate flexible supports is straightforward. A derivation, valid for the several types of splines, follows. The spline definition is given by Eq. 1-40, Eq. 1-50, and Eq. 1-54 and can be written as

Equation 1-55.

Equation 1-56. The structural deflection, ug , will differ from the spline deflection by the deformation of the spring, resulting in forces

Equation 1-57. where the diagonal matrix, Ks , has the spring constant, K, along the diagonal. These are nonzero (if K were equal to zero, then there would be no attachment and that grid point would be discarded) and thus the inverse of Ks is

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Equation 1-58. Eliminating uk between Eq. 1-56 and Eq. 1-57 and leads to

Equation 1-59. Thus, all that is required to accommodate springs is to add the spring flexibilities to the diagonal of the spline influence coefficient matrix. This is also obvious from physical reasoning, since the spring and spline flexibilities are in series and can be added directly.

Rigid Arms on Linear Splines The linear splines used for geometry interpolation have rigid arms (see Figure 1-2). Mathematically, these rigid arms represent equations of constraint between the displacements and rotations at the spline and the attachment point of the aerodynamic element. Physically, these rigid arms correspond to the assumption of a structure with a rigid chord perpendicular to the linear spline (elastic axis). The equations of constraint for the rigid arms are used to transform the influence functions from the spline ends to influence functions at the attachment points. The complete transformed influence functions are summarized at the end of this section.

The Constraint Spline The third spline in NX Nastran is simply a multipoint constraint useful for including control surfaces in aeroelastic analyses. The constraint has the form

Equation 1-60. where ud is the value of the dependent uk component, and ui is the displacement at grid Gi with component Ci .

Coordinate Systems and Constraints The spline constraints are derived in spline coordinates, transformed to the basic coordinate system, and then all splines are appended to a common constraint matrix, [Gkg ]. Finally, the multi-, single-, and omit-constraints are applied to reduce this to [Gka ].

Summary of Matrices for Spline Interpolation •

The Aij matrix in Eq. 1-56 for surface splines is

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Fundamentals of Aeroelastic Analysis

where: r2ij = (xi − xj )2 + (yi − yj )2 δij = and kz is the translational stiffness of an attached spring. •

The Ri matrix in Eq. 1-56 for surface and linear splines is

•

The Ai matrix in Eq. 1-56 for linear splines is

where kθ and kφ are rotational stiffnesses of the attached springs.

1.5

Static Aeroelasticity

Static (or quasi-steady) aeroelastic problems deal with the interaction of aerodynamic and structural forces on a flexible vehicle that results in a redistribution of the aerodynamic loading as a function of airspeed. The aerodynamic load redistribution and consequent internal structural load and stress redistributions are of concern to the structural analyst. The possibility of a static aeroelastic instability, (i.e., divergence) is also of concern to the structural analyst.

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The aerodynamic load redistribution and consequent modifications to aerodynamic stability and control derivatives are of interest to the aerodynamicist and the control systems analyst. The static aeroelastic capability in NX Nastran addresses these needs by the computation of aircraft trim conditions, with subsequent recovery of structural responses, aeroelastic stability derivatives, and static aeroelastic divergence dynamic pressures.

Generation of Aerodynamic Matrices The aerodynamic equations of Aerodynamic Theories form the basis of the aerodynamic computations required for static aeroelastic analysis with some special purpose modifications made for the NX Nastran implementation. For static aeroelasticity, the downwash relation of Eq. 1-2 becomes:

Equation 1-61. where: {wj } = vector of aerodynamic degrees of freedom (e.g., angles of attack) {uk } = vector of aerodynamic displacements (deformations) vector of “extra aerodynamic points” used to describe, e.g., aerodynamic control {ux } = surface deflections and overall rigid body motions represents an initial static aerodynamic downwash. It includes, primarily, the {wj g} = static incidence distribution that may arise from an initial angle of attack, camber, or washout (twist). substantial derivative matrix for the aerodynamic displacements. This is the Djk 1 [Djk ] = term of Eq. 1-2 and the Djk 2 term is not used for this quasi-steady analysis [Djx ] = substantial derivative matrix for the extra aerodynamic points The theoretical aerodynamic pressures are given by Eq. 1-20.

Equation 1-62. and the aerodynamic forces, based on Eq. 1-20 and Eq. 1-21, can be written

Equation 1-63. where all the terms have been defined in Aerodynamic Theories. The vector of aerodynamic extra points specifies the values of aerodynamic trim variables. NX Nastran has a number of predefined variables, including incidence angles (α and β), roll, pitch, and yaw rates (p, q, and r) and two translational (ü2 and ü3 ) and three rotational ( , and )

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accelerations. The DJX matrix, then, provides the vector of downwash velocities for unit values of these aerodynamic extra points. Note that accelerations do not result in any downwash for the quasi-steady assumption so that the corresponding columns of the DJX matrix are null.

Static Aeroelastic Equations of Motion The aerodynamic forces are transferred to the structure using the spline matrix in Eq. 1-22 and Eq. 1-25 reduced to the a-set to form an aerodynamic influence coefficient matrix, Qaa , which provides the forces at the structural grid points due to structural deformations

Equation 1-64. and a second matrix, Qax , which provides forces at the structural grid points due to unit deflections of the aerodynamic extra points:

Equation 1-65. The complete equations of motion in the a-set degrees of freedom require Kaa Maa Pa

Structural stiffness matrix Structural mass matrix Vector of applied loads (e.g., mechanical, thermal, and gravity loads plus aerodynamic terms due to user input pressures and/or downwash velocities)

The a-set equations are then:

Equation 1-66. This is the basic set of equations used for static aeroelastic analysis. In the general case, rigid body motions are included in the equations to represent the free-flying characteristic of an air vehicle. This is addressed in NX Nastran by a requirement that the user identify reference degrees of freedom equal in number to the number of rigid body motions using the SUPORT Bulk Data entry. Eq. 1-66 is then partitioned into r-set (supported) and l-set (left over) degrees of freedom yielding

Equation 1-67.

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where the notation

has been introduced. At this point, the NX Nastran implementation of aeroelastic analysis introduces a mathematical technique that is based on the NX Nastran inertia relief analysis without aeroelastic effects. The technique entails multiplying the first row of Eq. 1-67 by DT and adding the result to the second row, where

Equation 1-68. is known as the rigid body mode matrix and can be shown to be only a function of the geometry of the model. The resulting set of equations is then

Equation 1-69. If there were no aerodynamic terms, the DTKll + Krl and DTKrl + Krr would sum to zero so that the second row of equations could be solved for {ür }. With the aerodynamic coupling, this simplification is not possible. DTKll + Krl = 0 from the definition of D given in Eq. 1-68. Note that DTKlr + Krr = 0 because this represents the work performed on the structure when it undergoes a rigid body displacement. It is seen that Eq. 1-69 contains nl + nr equations with 2(nl + nr) undetermined quantities, where nl, nr, and nx are the number of degrees of freedom in the l and r sets and the number of aerodynamic extra points, respectively. The undetermined accelerations can be directly specified using two relations. The first relation comes from the assumption of quasi-steady equilibrium and specifies that

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Equation 1-70. where [D] is the rigid body mode matrix of Eq. 1-68, and Eq. 1-69 simplifies to

Equation 1-71. where [mr] = [Mrr + Mrl D + DTMlr + DTMll D] is the “total” mass matrix relative to the ur points. The second relation recognizes that the {ür } structural accelerations are related to the aerodynamic extra points {ux } via

Equation 1-72. where TRX is a Boolean matrix that selects accelerations from the aerodynamic extra points and [TR]T is a matrix that transforms accelerations from the aerodynamic reference point to the “supported” degrees of freedom. This second matrix is a function of only the geometry of the model. The further solution of the static aeroelastic equations is dependent on the type of analysis required. The remainder of the section is divided into four subsections that treat (1) restrained analysis for trim and stability derivative analysis, (2) unrestrained stability derivative analysis, (3) rigid stability derivatives, and (4) divergence analysis.

Restrained Analysis Significant simplification is made by assuming that the {ur } terms can be set to zero with the remaining displacements then computed relative to this assumption. Therefore, setting ur = 0 in Eq. 1-71, and solving for ul from the first row gives:

Equation 1-73. This is then substituted into the second row of Eq. 1-71 and the relationship for {ür } in terms of {ux } of Eq. 1-72 is used to give nr equations with only the ux quantities undetermined:

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Equation 1-74. where:

Equation 1-75.

Equation 1-76. The solution of Eq. 1-74 for ux requires that the equation be augmented by user input relations that specify all but nr terms in the ux vector. These user specifications can be done by either specifying ux values directly or by specifying a linkage that makes a term (or terms) in the ux vector dependent on an independently varying term. The augmented trim equation then has the form:

Equation 1-77. where IP is a pseudo-identity matrix with as many rows as there are user-specified constraints on the values of ux terms. The IP matrix has ones in the row and columns corresponding to the constrained variables and zero elsewhere. The Y vector contains the values of the user-specified constraints. The AEL matrix contains any user-specified relationships between (or among) aerodynamic extra points. Eq. 1-77 can then be solved for the remaining terms in the ux vector. Once the ux vector has been evaluated, {ül } and {ul } can be recovered using Eq. 1-72, and Eq. 1-73, respectively. The software uses standard NX Nastran data recovery techniques to compute user-requested values of displacements, stresses, etc. The user may also request pressures and forces on the aerodynamic boxes or elements. These terms are calculated using Eq. 1-62 for the pressures and Eq. 1-63 for the forces. Stability derivatives are also calculated using the ZZX and PZ matrices with some modifications. For stability derivatives associated with aerodynamic extra points, dimensional stability derivatives are obtained from

Equation 1-78.

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Stability Derivatives The second term on the right-hand side represents removing the vehicle accelerations from the ZZX matrix. Nondimensional stability derivatives are then calculated using:

Equation 1-79. where S is the reference area of the vehicle, TR transforms forces from the support location to the aerodynamic reference point and

Equation 1-80. where cref and bref are the reference chord and span, respectively. The intercept coefficients are computed using only partitions of the PZ vector that are dependent on , i.e., terms associated with user input downwashes and user input pressure coefficients. This means that applied mechanical, thermal, and gravity loads are neglected in the following calculation:

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Equation 1-81.

Unrestrained Stability Derivatives The stability derivatives of the previous section were computed under the assumption of {ur } = 0. This calculates these quantities in a structural axis system that is dependent on the arbitrary selection of the support point location. The stability derivatives of an unrestrained vehicle must be invariant with the selection of the support point location. This invariance is obtained by introducing a mean axis system. The characteristic of the mean axis is that deformations of the structure about it occur such that there is neither movement of the center of gravity nor rotation of the principal axes of inertia. Stated another way, the displacements in the mean axis system are orthogonal to the rigid body modes of the vehicle. In terms of the rigid body mode matrix [D] and the mass matrix of the system [Maa ], the mean axis constraint is defined by

Equation 1-82. or

Equation 1-83. Eq. 1-71 and Eq. 1-83 can be combined to give an overall system of equations:

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Fundamentals of Aeroelastic Analysis

Chapter 1

Equation 1-84. The first row of this equation can be solved for ul in terms of ur and ür and ux

Equation 1-85. where the ARLR, AMLR, ALX, and UINTL terms can be inferred and the notation used conforms to that used in the static aeroelastic subDMAP (AESTATRS) described in Flutter Analysis Sample Problems. The INTL vector is the aerodynamic related portion of the Pl vector. The expression for ul of Eq. 1-85 can be placed in the second row of Eq. 1-84 to give

Equation 1-86. where: [M2RR] =

[DTMlr + Mrr ] − [DTMll + Mrl ][ARLR]

[M2RR] =

−[DTMll + Mrl ][AMLR]

[K3LX]

=

−[DTMll + Mrl ][ALX]

[TMP1]

=

[DTMll + Mrl ]{UINTL}

Eq. 1-86 is solved for ur in terms of ür and ux to give:

Equation 1-87. where:

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[M4RR]

=

[M2RR]−1[M3RR]

[K4LX]

=

[M2RR]−1[K3LX]

{TMP2}

=

[M2RR]−1 {TMP1}

If the expression for ul from Eq. 1-85 is placed in the third row of Eq. 1-84, then:

Equation 1-88. where: [K2RR]

=

−[DTKall + Karl ][ARLR] + [DTKalr + Karr ]

MSRR

=

mr

KAZL

=

DTKall + Karl

KARZX

=

KAZL − KAXL · ALX

{IPZ}

=

{INTZ} − {DTKall + Karl }{UINTL}

where {INTZ} is the aerodynamic portion of {DTPl + Pr } Next, the ur expression of Eq. 1-87 is placed in Eq. 1-88 to give:

Equation 1-89. where: [M5RR]

=

−[K2RR][M4RR] + [MSRR]

[MIRR]

=

−[KAZL][AMLR] + [M5RR]

[KR1ZX]

=

−[K2RR][K4LX] + [KARZX]

{IPZF}

=

[K2RR]{TMP2} + {IPZ}

The stability derivatives require an equation that premultiplies the ür term by the rigid body mass matrix. This is achieved by premultiplying Eq. 1-89 by [MSRR][MIRR]−1

Equation 1-90. where: {IPZF1}

=

[MIRR]−1 {IPZF}

{IPZF2}

=

[MSRR]{IPZF1}

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Fundamentals of Aeroelastic Analysis

Chapter 1

[KR2ZX]

=

−[MIRR]−1 [KR1ZX]

[Z1ZX]

=

[MSRR][K2RZX]

In a manner similar to the restrained case, the unrestrained stability derivatives can be obtained using

Equation 1-91. Rigid Stability Derivatives and Mean Axis Rotations This subsection briefly provides a theoretical description of several data blocks that are used to provide output to the user. Two sets of rigid stability derivatives are printed out. These are stability derivatives that are computed when elastic deformations are neglected. The first set of rigid derivatives are denoted as “unsplined” and are the values obtained directly from the aerodynamic calculations before they have been transferred to the structure. The dimensional matrix that contains these derivatives is

where:

[see Eq. 1-65], and SRKT is a matrix that sums forces acting on each of the aerodynamic boxes or elements to the supported degrees of freedom. This matrix is only a function of the geometry of the aerodynamic model and the locations of the support degrees of freedom.

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The intercept stability dimensional derivatives are computed using

Equation 1-92. Similar expressions are available to the splined rigid stability derivatives and intercepts

Equation 1-93.

Equation 1-94. Nondimensionalization of these matrices is performed in a fashion similar to that given in Eq. 1-79 for stability derivatives and Eq. 1-81 for intercept values. The availability of these rigid terms is useful in several ways. A comparison between the splined and unsplined derivatives provides an assessment of the quality of the splining. If the numbers differ significantly, this may indicate that not all aerodynamic elements have been transferred to the structure. This may be the user’s intent or it may indicate a user error. If the two sets of rigid numbers bear little resemblance to one another, a serious splining error has been made. The guidelines of Aeroelastic Modeling should be consulted to determine if there is a modeling error. Similarly, experience should allow the user to assess the reasonableness of the flexible results when compared with the rigid numbers. Large differences indicate large structural deformations and may point up conditions such as local weaknesses in the structure, an aerodynamic model displaced from the structural model, or errors in the input of the flight condition. The rotations of the mean axes relative to the structural axes through the support points are required when restrained aeroelastic coefficients (stability derivatives and intercept coefficients) are used in the equations of motion [see Rodden and Love (1985)]. The deflections of the mean axes relative to the origin of the structural axes are derived as follows. The grid point deflections relative to the mean axes u¯ l are related to the deflections of the SUPORT points u¯ r through the rigid body mode matrix [D] (see Figure 1-3).

Equation 1-95. The requirement for the mean axes is that deformation occurs about them such that the center of gravity does not move and the axes do not rotate. In terms of the rigid body mode matrix and the mass matrix, this condition is expressed by

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Chapter 1

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Equation 1-96. Eq. 1-95 and Eq. 1-96 lead to

Equation 1-97. Introducing Eq. 1-73 yields

Equation 1-98. and with Eq. 1-72 we finally obtain

Equation 1-99. The coefficient of {ux } in Eq. 1-99 is defined as the matrix [HP] corresponding to the aerodynamic extra points

The second term in Eq. 1-99 is defined as the matrix [HP0] corresponding to the user inputs of {wgj } and

.

The columns of deflections in [HP] are used to find the mean axis rotations for each aerodynamic extra point, αmi in the longitudinal case, βmi in the lateral case, and γmi in the directional case, for the user inputs; [HP0] is used in like manner to find the longitudinal mean axis rotation αm0 . The general problem of obtaining the mean axis rotations with multiple SUPORT points is a problem in solid analytical geometry that is beyond the scope of this guide, but is also not regarded as a practical situation. If there is only a clamped SUPORT at one point (see Figure 1-3) the deflections in [HP] and [HP0] are not needed; only the rotations are of interest, and these are illustrated in the examples of Static Aeroelastic Analysis Sample Problems. In the

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more general longitudinal case of two SUPORTed grid points (see Figure 1-4) the longitudinal mean axis rotations αmi and αm0 are found by dividing the difference between the upstream and downstream deflections by the distance between the two grid points. Note that “upstream” and “downstream” must be determined by PARAM,USETPRT,11 in the Bulk Data to account for any resequencing of grid points in the NX Nastran solution.

Figure 1-3. Geometry of Deformed Flight Vehicle with Clamped SUPORT

Figure 1-4. Geometry of Deformed Flight Vehicle with Multiple SUPORTs

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Speed Derivatives NX Nastran assumes a constant forward velocity in all of its aeroelastic solutions and has no capability to estimate induced drag coefficients. However, certain speed derivatives that are required in longitudinal maneuvering studies of aircraft can be obtained using NX Nastran. The lift and moment aeroelastic coefficients are functions of two speed parameters: Mach number and dynamic pressure, so the expression for the speed derivative is

The numerical evaluation of the derivative; e.g., with respect to Mach number, is done with the finite difference

In the calculation of the finite difference derivatives, the perturbations, e.g., Δm, must be large enough that the close differences do not lose significance. The above example of the finite difference derivative is a forward difference; a central difference formula might also be used using perturbations on either side of the flight condition. Three NX Nastran subcases are required to evaluate the speed derivatives by the forward difference equation: •

at the flight condition of m and

•

with a perturbed Mach number m + Δmand

•

with the flight Mach number m and a perturbed dynamic pressure

Divergence Analysis The divergence speeds of a restrained aircraft component may be obtained by solving an eigenvalue problem. The divergence eigenvalue problem for the restrained vehicle can be extracted from the (1,1) partition of Eq. 1-71 above:

Equation 1-100. Since

, the divergence eigenvalue problem becomes

Equation 1-101. where the eigenvalues are the dynamic pressures for divergence. Only positive values of have any physical significance and the lowest value of is the critical divergence dynamic pressure.

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1.6

Flutter Solution Techniques

Flutter is the dynamic aeroelastic stability problem. It can be solved in any speed regime simply by selecting the appropriate aerodynamic theory. In the linear case assumed throughout this guide, the solution involves a series of complex eigenvalue solutions; the eigenvalue problem to be solved depends on the way in which the aerodynamic loads are included in the equations of motion or whether certain damping terms are included. The manner in which the aerodynamic loads are included depends on how the dimensionless oscillatory aerodynamic coefficients are defined. When Theodorsen (1935) first developed the American method (K-method) of flutter analysis, he introduced the aerodynamics into a vibration analysis as complex inertial terms and the flutter analysis became a vibration analysis requiring complex arithmetic. At the same time, he introduced an artificial complex structural damping, proportional to the stiffness, to sustain the assumed harmonic motion. Flutter analysis is then a double eigenvalue problem in frequency and velocity, and an iterative solution, using the reduced frequency of the assumed harmonic motion as the iteration parameter, leads to the neutrally stable conditions (flutter frequencies and velocities) at which no artificial damping is required. The artificial damping is therefore seen not to be physically meaningful, other than, perhaps, at speeds near flutter speeds. At about the same time, Frazer and Duncan (1928) in England were attempting to solve the flutter problem using aerodynamic stability derivatives in the tradition of Bryan (1911) who had studied the flight mechanics of rigid aircraft. This approach introduced the aerodynamic loads into the equations of motion as frequency dependent stiffness and damping terms. In this representation it should be noted that the aerodynamic terms are slowly varying functions of the reduced frequency, in contrast to the representation of the aerodynamics in the K-method as mass terms that are highly dependent on the reduced frequency. In what has become known as the “British” method of flutter analysis some iteration is still necessary to “line-up” the eigenvalue solution for frequency with the reduced frequency in each mode. A description of the British method and a comparison with the American method has been given by Lawrence and Jackson (1970). A variation of the British method in which the aerodynamic loads are treated as complex springs has been developed by Hassig (1971). Hassig called his method the p-k method, and NX Nastran has adopted his terminology, although it is now applied to the British method. The NX Nastran terminology is K-method for the American method, and PK-method for the British method. NX Nastran also has a very efficient K-method, called the KE-method, but it does not provide eigenvectors and has no provisions for viscous damping type terms, such as arise in an automatic control system in the equations of motion.

Generalized Aerodynamic Matrices Eq. 1-4 defines an aerodynamic influence coefficient matrix Qkk that is computed based on the aerodynamic model. For this matrix to be useful in a flutter analysis, two transformations have to take place: 1. The matrices have to be applied to the structural model using the spline techniques discussed in Aeroelastic Modeling. 2. A modal reduction has to be applied to obtain the matrices in generalized form. Mathematically, those transformations can be expressed as

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Equation 1-102. where: Qii φai Gka WTFACT

= = = =

the generalized aerodynamic matrix a matrix of i-set normal mode vectors in the physical a-set the spline matrix of Eq. 1-22 reduced to the a-set the same weighting matrix as Wkk defined following Eq. 1-21

A level of complexity is added if the flutter analysis includes the use of extra points. Extra points are used for the representation of control systems and are therefore required in aeroservoelastic analyses. If the extra point deflections result in displacements of the aerodynamic model, the user is required to provide the downwash information explicitly on two DMI entries:

Equation 1-103. where [ue ] is a vector of extra point displacements, D1JE and D2JE are the required matrices, and wj is the resulting downwash. Generalized aerodynamics for these extra points can be computed as

Equation 1-104. where, similar to Eq. 1-4,

Equation 1-105. The flutter analysis then uses a merged matrix:

Equation 1-106. where the h-set is a combination of the i-set normal modes and the e-set extra points. It is seen that the lower e-set rows in the matrix are null. Physically, this indicates that the normal mode deflections do not produce aerodynamic forces on the extra points (Qei = 0) and that the extra point deflections do not produce aerodynamic loads on the extra points (Qee = 0).

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Interpolation of Qhh In a typical flutter analysis, the computation of the aerodynamic matrices and the subsequent processing of these matrices to generate the generalized aerodynamic matrices of the previous subsection represents a significant portion of the computer resources required to perform the analysis. These matrices are generated for discrete values of Mach number and reduced frequency. As detailed in the flutter algorithm discussions to follow, the actual flutter analysis is likely to be performed at reduced frequencies (and sometimes Mach numbers) other than one of the available values. These intermediate values are obtained from an interpolation of the available values, and three separate interpolation schemes have been implemented in NX Nastran. The first two are available for the K- and KE-flutter methods and are special applications of the linear and surface spline. The third is used with the PK-flutter method and is a linear spline method that has been further tailored to aerodynamic applications. For the purpose of this discussion, the three methods are designated (1) linear spline, (2) surface spline, and (3) special linear spline.

Linear Spline The linear spline of Interconnection of the Structure with Aerodynamics is applied to the aerodynamic interpolation task with the spline axis representing reduced frequency values. The torsion terms are ignored in this case, as are the options of attaching springs. Under these circumstances, the interpolation can be written as:

Equation 1-107. where Cj is determined from

Equation 1-108. and [A] is a symmetric nhdpts + 2 matrix with

Equation 1-109.

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Equation 1-110. Note that only the first nhdpts rows of the C vector are required in performing the interpolation.

Surface Spline The surface spline of can be applied in the aerodynamic interpolation task with reduced frequencies and Mach numbers taking the place of the x- and y-coordinates for this application. Again, a number of simplifications can be made for this application, allowing for a terse description of the interpolation algorithm. Eq. 1-107 and Eq. 1-108 can still be used to represent the interpolation, A is now a symmetric nhdpts + 3 matrix with:

Equation 1-111.

Equation 1-112. where:

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Equation 1-113. As in the case of the linear spline, only the first nhdpts rows of the C vector [Eq. 1-107] are required to perform the interpolation.

Special Linear Interpolation The interpolation task can be thought of as a relatively minor task for the K- and KE-methods in that the user defines the hard points and the interpolated points and can be expected to make a reasonable selection. As detailed below, the PK-method computes the reduced frequency values to which aerodynamics are to be interpolated without user intervention. This calls for a more robust interpolation scheme. For the PK-method, the interpolation has the form

Equation 1-114. where QIhh /kj is fit rather than Qhh I directly since the former quantity is a smoother value of k and because it is needed in the formulation of Eq. 1-128. Further, this makes the response data symmetric with respect to a reduced frequency of zero and this “boundary condition” can be applied in the interpolation. The C vector of Eq. 1-114 is

Equation 1-115. with

Equation 1-116.

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Equation 1-117. where the ki = kj and kest = kj terms result from the symmetry condition.

The K-Method of Flutter Solution The basic equation for modal flutter analysis by the K-method is

Equation 1-118. where: Mhh

=

modal mass matrix, usually (but not necessarily) diagonal

Bhh

=

modal damping matri

Khh

=

modal stiffness matrix, usually (but not necessarily) diagonal; may be complex (with actual structural damping); will be singular if there are rigid body modes

m

=

Mach number

k

=

reduced frequency =

=

reference length

Qhh (m,k)

=

aerodynamic force matrix, which is a function of parameters m and k

ω

=

circular frequency = 2πf

g

=

artificial structural damping

ρ

=

fluid density

V

=

velocity

uh

=

modal amplitude vector, sometimes called modal participation factors

Note that k, V, and ω are not independent. For the K-method of solution, the aerodynamic term is converted to an equivalent aerodynamic mass

Equation 1-119.

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for mathematical convenience, The term involving Bhh in Eq. 1-119 has been multiplied by and is valid only at flutter, i.e., when g = 0. Eq. 1-119 is solved as an eigenvalue problem for a series of values for parameters m, k, and ρ. The complex eigenvalue is ω2/(1 + ig), which can . Flutter be interpreted as real values of ω and g. The velocity, V, is recovered from occurs for values of m, k, and ρ for which g = 0. The solutions are not valid except when g = 0, since the aerodynamic force terms are valid only for sinusoidal motion and g is not a physical damping. A slight variation has been used with NX Nastran. The equation is written as

Equation 1-120. Thus, the new eigenvalue is

Equation 1-121. Eq. 1-121 may be written as

Equation 1-122. so that

Equation 1-123.

Equation 1-124.

Equation 1-125.

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The eigenvalue problem is expressed in a quadratic form in Eq. 1-111. The manner of solution depends on which of the various complex eigenvalue methods available in NX Nastran is selected by the user. For more information on the theory of complex eigenvalue analysis, see: •

Theory of Complex Eigenvalue Analysis in the NX Nastran Numerical Methods User’s Guide

The K-method of flutter analysis is a looping procedure. The values of V, g, and f = ω/2π are solved for various values of m, k, and ρ. Plots of V versus g can be used to determine the flutter speed(s) (where g goes through zero to positive values).

The KE-method of Flutter Solution A more efficient K-method of flutter analysis is possible if the analyst is willing to neglect viscous dampings from all sources, e.g., from the structure or a control system, and to restrict the solution to eigenvalues and not require eigenvectors. The equation to be solved becomes Eq. 1-120 with the term containing Bhh deleted; note that complex structural damping may still be included in Khh . Many of the operations can then be done in-core with a consequent increase in efficiency. This efficient K-method algorithm is called the KE-method. With this increase in efficiency, a greater number of points on a flutter stability curve can be obtained for a given cost, and cases with poorly behaved stability curves can be studied more thoroughly. This method gives results similar to those of Desmarais and Bennett (1974). In order to sort the roots so that curves can be drawn, the roots must be ordered. For the first value of k, the roots are accepted in the order output by the upper Hessenberg eigenvalue subroutine. If we denote the i-th eigenvalue for the n-th reduced frequency kn by pi, n, we may define an extrapolated eigenvalue as

Equation 1-126. in which pi, 0 is chosen equal to pi, 1 . Then the values of pi, n are ordered according to closeness to pi, n(e) where the “closeness” is measured by a minimum value of

Equation 1-127. With this sorting, the V-g and V-f curves produced using the NX Nastran NASPLOT utility can be interpreted physically. The curves from the K-method, on the other hand, are extremely difficult to interpret.

The PK-Method of Flutter Solution The fundamental equation for modal flutter analysis by the PK-method is

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Equation 1-128. where the new terms are: modal aerodynamic damping matrix, a function of Mach number, m, and reduced frequency, k modal aerodynamic stiffness matrix, a function of Mach number, m, and QRhh = reduced frequency, k p = eigenvalue = ω(γ ± i) γ = transient decay rate coefficient (Note that the structural damping coefficient g = 2γ) QIhh =

The matrix terms in Eq. 1-128 are all real. QRhh and QIhh are, respectively, the real and imaginary parts of Qhh (m,k). Note that the circular frequency and the reduced frequency are not , and furthermore, that independent since

Equation 1-129. For the PK-method of solution, Eq. 1-128 is rewritten in the state-space form with twice the order.

Equation 1-130. where [A] is the real matrix

Equation 1-131. and {u¯ h } now includes both modal displacements and velocities. The eigenvalues of the real matrix [A]] are either real or complex conjugate pairs. Real roots indicate a convergence or divergence as in the cases of the roll subsidence (rigid body) mode or a structural (torsional) divergence mode. For the real roots, the damping is expressed as the decay rate coefficient, which is the distance travelled (measured in chord lengths) to half (or double) amplitude.

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Equation 1-132. However, the majority of the eigenvalues will be complex conjugate pairs. The oscillatory solutions of Eq. 1-128 require an iterative solution so that Eq. 1-129 is satisfied along with Eq. 1-128. The roll subsidence root or the static structural divergence roots require no iteration but are found by setting k = 0. The oscillatory rigid body (i.e., short-period or Dutch-roll) roots and oscillatory roots are found from the following algorithm. This algorithm is based on a desire for capability to determine stability at a given speed independently of the stability at lower or higher speeds and the fact that the aerodynamic terms in Eq. 1-128 are slowly varying functions of the reduced frequency. The iteration begins at k = 0 [QRhh and QIhh /k are extrapolated to k = 0 from the available values of Qhh (m,k)] and all real roots immediately satisfy Eq. 1-129 but the complex roots do not. The iteration for the complex roots then proceeds as follows. In general, let the complex pairs of eigenvalues be written as

Equation 1-133. where r denotes the oscillatory mode number ordered by frequency (ω1s < ω2s < ...), s denotes the number of the oscillatory mode under investigation, and j denotes the iteration (eigenvalue solution) number so that the next estimate of the (nonzero) reduced frequency is

To find the first oscillatory root the estimate of the first nonzero reduced frequency is taken as

Equation 1-134. Convergence to the first oscillatory root then occurs when

Equation 1-135. where ε is a user input with a default value of 0.001. Let the converged complex eigenvalues be

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Equation 1-136. where only pss (c) satisfies both Eq. 1-129 and Eq. 1-131. Then the search for the next oscillatory mode begins by increasing s by one, and the first estimate of the next reduced frequency is

Equation 1-137. The iteration of Eq. 1-133 and Eq. 1-134 continues until

Equation 1-138. is satisfied. Eq. 1-136 and Eq. 1-137 begin the search for each of the higher modes of interest. The eigenvalues and eigenvectors of Eq. 1-129 are found by a real version of the Upper Hessenberg method described by Komzsik For more information, see: •

Hessenberg Method in the NX Nastran Numerical Methods User’s Guide

The order of calculations for different values of density, velocity and/or Mach number is indicated in Flutter Analysis. The principal advantage of the PK-method is that it produces results directly for given values of velocity, whereas the K- and KE-methods require iteration to determine the reduced frequency of flutter. In addition, the damping given by 2γss (c)found from Eq. 1-136 is a more realistic estimate of the physical damping than the parameter g in Eq. 1-132, which is a mathematical artifice.

1.7

Dynamic Aeroelastic Analysis

Dynamic aeroelasticity differs from the flutter analysis of the previous section in that the right-hand side of Eq. 1-118 is no longer zero. Instead, loading, which can be in either the frequency or the time domain, is applied. For both types of loading, NX Nastran performs the primary analyses in the frequency domain. If the user has supplied loadings in the time domain, Fourier Transform techniques are used to convert the loadings into the frequency domain, a frequency response analysis is performed and the computed quantities are transformed back to the time domain using Inverse Fourier Transform techniques. This section first describes the frequency response analysis that is the basis of all NX Nastran dynamic aeroelastic analysis and then discusses the special topics of transient response analysis and random response analysis.

Aeroelastic Frequency Response Analysis Aeroelastic frequency response analysis in NX Nastran is performed in modal coordinates and has a basic equation of the form

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Equation 1-139. where all terms on the left-hand side are identical to those of Eq. 1-118 and are defined with that equation. The right-hand side provides the loading in modal coordinates, which can be aerodynamic or nonaerodynamic in nature and is a function of the analysis frequency. Nonaerodynamic generalized loads, designated PHF(ω), are obtained in the standard fashion from the loadings applied to physical coordinates and do not require further comment here. The aeroelastic (gust) portion of the loading does require further comment that is similar in nature to the discussion of the generalized aerodynamic matrices of the previous subsection. A prerequisite to performing aerodynamic gust analysis is the availability of an aerodynamic matrix that provides the forces on the aerodynamic elements due to an applied downwash at any other element:

which can be transformed to modal coordinates using:

Since extra points cannot affect the gust loading, there are no generalized loadings associated with them so that matrix Qhj (which provides the generalized loadings in the modal set) is obtained by adding a null matrix onto the bottom of Qij . The Qhj matrix supplies the generalized aerodynamic forces due to the downwash vector at the collocation points. For the matrix to be useful in the gust analysis, two other steps are required. First, the Qhj and Qhh matrices must be interpolated to all the frequencies required in the analysis from the discrete reduced frequencies at which the aerodynamics have been calculated. This is done using the Specialized Linear Interpolation technique of the previous subsection applied to the two matrices. The other step is the generation of the gust downwash matrix. This is a function of frequency and the geometry of the aerodynamic model:

Equation 1-140. where: ωi γj xj xo

= = = =

excitation frequency dihedral angle of the j-th aerodynamic element x-location of the j-th aerodynamic element in the aerodynamic coordinate system user-supplied offset distance for the gust

It is seen that this represents a one-dimensional gust field, i.e., the gust varies only in its x-coordinate. The generalized load due to the aerodynamic gust is then

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Equation 1-141. where: = dynamic pressure wg = gust scale factor user-supplied frequency variation of the gust. (This may also be obtained from a Fourier PP = transform of the user-supplied discrete gust.) and the total frequency dependent loading applied in Eq. 1-139 is

Equation 1-142. The solution of Eq. 1-139 entails solving for the generalized displacements by decomposition/forward-backward substitution techniques applied to the coupled set of complex equations. Because modal reduction techniques have been applied, the solution costs are typically modest. Once the generalized displacements have been computed, standard data recovery techniques can be used to determine physical displacements, velocities, stress, etc.

Aeroelastic Transient Response Analysis As discussed in the introduction to this section, Aeroelastic Transient Analysis relies on Fourier transform techniques. Transient analysis by a Fourier transformation is separated into three phases. First, the loads (defined as a function of time) are transformed into the frequency domain. Second, the responses are computed in the frequency domain using the algorithm of the preceding subsection. Third, these responses (in the frequency domain) are transformed back to the time domain. Two forms of the transform are considered, the Fourier series and the Fourier integral, which are defined using the following terminology. The Fourier Series The basic time interval is 0 < t < T, with the function periodic. The circular frequencies are given by

Equation 1-143.

Equation 1-144. The load transformation for a load at point a is

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Equation 1-145. The response at point j is given by

Equation 1-146. The response in the time domain is

Equation 1-147. The Fourier Integral This is the limit as T → ∞, Δf → 0, with 2πnΔf→ ∞, of the Fourier series. Here, ω is a continuous variable.

Equation 1-148.

Equation 1-149.

Equation 1-150. Both methods must be implemented numerically, which requires approximations that the user should appreciate. First, the inverse transform includes an infinite sum, for which only a finite number of terms are evaluated numerically; this approximation leads to truncation errors. Next, the inverse Fourier integral must be integrated numerically, which leads to integration errors. Finally, the number of frequencies at which the integrand is evaluated is limited by the cost of calculations. Each of these sources of error should be addressed separately by the user to ensure adequate accuracy in the final results.

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Transformation of Loads You specify loads in the same manner as given in Transient Excitation Definition in the NX Nastran Basic Dynamic Analysis User’s Guide . The two general forms are the tabular, piecewise linear function [Eq. 1-151 below] and the general purpose function [Eq. 1-156 below]. The transformation is given by Eq. 1-148, in which it is assumed that the user defines a function which vanishes for t > T. For piecewise linear tabular functions, a table of pairs xi , yi , i = 1, Nis prescribed, which defines N− 1 time intervals. In addition, an X1 shift and an X2 scale factor are allowed. Thus, the time-dependent load at point j is given by

Equation 1-151. where YT is the tabular function supplied by the user-defined by the N pairs (xi , yi ), and Aj and tj are an amplitude factor and a delay, respectively, which may depend upon the point which is loaded. The transformed load is

Equation 1-152. with

Equation 1-153.

Equation 1-154. and

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Equation 1-155. The general purpose function is defined by

Equation 1-156. where:

Equation 1-157. The value of n is restricted to be an integer for transient analysis by the Fourier method. The transform is given by

Equation 1-158. where:

Equation 1-159.

Equation 1-160.

Equation 1-161. and R1 is the same as R2 except the signs of φ and f are reversed. Also,

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Equation 1-162. The second form is used for |z| < 0.1. These loads, which are in the form required for frequency response, are transformed to the modal coordinates exactly as in the modal frequency response method. Inverse Transformation of the Response The response is found from a numerical approximation to Eq. 1-146 or from Eq. 1-143, the Fourier series result, which can be regarded as a special form of approximation to the integral. The quantity u(ω) ˜ is first calculated at a set of frequencies, ωi , by the frequency response analysis. The ωi do not need to be equally spaced and the integral is evaluated only over the frequency range for which the frequency response has been performed. The frequency response is taken to be constant over the frequency interval resulting in a discrete inverse transform of the form:

Equation 1-163. where Re denotes the real part and C1 = Cn = CNFREQ = (ωNFREQ − ωNFREQ − 1 )/2 A special case occurs if all the frequency intervals are equal to Δf and the first frequency is an integer multiple of Δf. In this case, the time step Δt is adjusted to make 1/ΔfΔt = an integer, thereby reducing the number of distinct values of sinωn tm and cosωn tm used in Eq. 1-163, and cn does not require recalculation at each frequency. Practical Considerations for Use with the Fourier Methods Some important practical considerations must be observed to use these methods successfully. To illustrate one problem, consider the response of a simple damped oscillator to a pulse (Figure 1-5). The upper three curves show the pulse and the response of the system, first, if it is very stable, and then if it is only slightly stable. Using the Fourier method, the pulse is replaced by a

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series of pulses with period 1/(Δf). As can be seen, this gives an accurate representation if the system is very stable, but an incorrect impression if the system is only slightly stable. Guidelines that lead to valid results using the Fourier method include: 1. The system should be stable. 2. The forcing functions should be zero for some time interval to allow decay. 3. The frequency interval Δf ≤ 1/(Tpulse + Tdecay )

Figure 1-5. Response of a Damped Oscillator to a Triangular Pulse If the system has unstable modes, these will appear as a precursor before the pulse, just like a “stable mode in reverse time.” The use of both equal frequency intervals and unequal intervals has been studied briefly and results are shown by Rodden, Harder, and Bellinger (1979) for a lightly damped, single degree-of-freedom oscillator. It has been found there that the combination of a few well-chosen values near the resonant frequencies and a uniformly spaced set of frequencies elsewhere produces reasonable results for the lightly damped example considered. However, further convergence studies on more general examples are needed.

Random Response Analysis The major loads to which an aerospace vehicle is subjected can be predicted for the most part from its design mission and maneuvering requirements. However, the total environment cannot be predicted exactly and statistical methods based on the theory of random processes must be employed to complete the description. Examples of random processes in aeroelasticity include response to atmospheric gusts and to aerodynamic buffeting. The random process theory considered in NX Nastran is based on generalized harmonic analysis, i.e., frequency response techniques, and assumes that the system is linear and that both the excitation and response are stationary with respect to time. A fundamental quantity in random analysis theory is the autocorrelation, Rj (τ), of a physical variable, uj , which is defined by

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Equation 1-164. Note that Rj (0) is the time average value of uj 2, which is an important quantity in the analysis of structural fatigue failure. The power spectral density Sj (ω) of uj is defined by

Equation 1-165. It may be shown (using the theory of Fourier integrals) that the autocorrelation function and the power spectral density are Fourier transforms of each other. Thus

Equation 1-166. from which the mean-square theorem follows

Equation 1-167. The factor 1/(2π) in Eq. 1-166 is omitted by some authors, or is sometimes replaced by other factors. The value of the factor depends on the definition of Sj (ω) Eq. 1-165. The expected value of the number of zero crossings with positive slope per unit time, or mean frequency, is another quantity of interest for fatigue analysis and design of aircraft for gusts. This mean frequency, N0 , can be found from the power spectral density:

Equation 1-168. The mean frequency, N0 , is thus the root mean square frequency, in which the power spectral density is used as a weighting function.

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The transfer function theorem [see, e.g., Bisplinghoff, Ashley, and Halfman (1955, App. C)] states that if Hja (ω) is the frequency response of any physical variable, uj , due to some excitation source, Qa (t), (e.g., a point force, or a distributed loading condition), so that if

Equation 1-169. where uj (ω) and Qa (ω) are the Fourier transforms of uj (t) and Qa (t), then the power spectral density of the response, Sj (ω), is related to the power spectral density of the source, Sa (ω), by

Equation 1-170. Eq. 1-170 is an important result because it allows the statistical properties (e.g., the autocorrelation function) of the response of a system to random excitation to be evaluated via the techniques of frequency response. If the cross-correlation function between any pair of sources

Equation 1-171. is null, the sources are said to be statistically independent and the power spectral density of the total response is equal to the sum of the power spectral densities of the responses due to individual sources. Thus,

Equation 1-172. If the sources are statistically correlated, the degree of correlation can be expressed by a cross-spectral density, Sab , and the spectral density of the response may be evaluated from

Equation 1-173. where Hjb * is the complex conjugate of Hjb . Note that [Sab ] is a Hermitian matrix, i.e., [Sab ]T = Sab *. Eq. 1-173 can be generalized for multiple responses to

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Equation 1-174. In applying the theory, it may not be necessary to consider the sources to be forces at individual points. An ensemble of applied forces that is completely correlated (i.e., a loading condition) may also be treated as a source. For example, a plane pressure wave from a specified direction may be treated as a source. Furthermore, the response may be any physical variable including internal forces and stresses as well as displacements, velocities, and accelerations. In NX Nastran, random response analysis is treated as a data recovery procedure that is applied to the results of a frequency response analysis. The frequency response analysis is performed for loading conditions, {Pa }, at a sequence of frequencies, ωi . Normal recovery procedures are applied to the output of the frequency response analysis module [see Random Response Analysis in the NX Nastran Advanced Dynamic Analysis User’s Guide], resulting in a set of output quantities, uj . The calculation of power spectral densities and autocorrelation functions for the output quantities is performed in the random response analysis module. The inputs to the calculation are the frequency responses, Hja (ωi ), of quantities uj to loading conditions {Pa } at frequencies ωi and the auto- or cross-spectral densities of the loading conditions, Sa or Sab . The response quantities, uj may be displacements, velocities, accelerations, internal forces, or stresses. The power spectral densities of the response quantities are calculated by Eq. 1-173 or by Eq. 1-170, depending on whether the loading conditions are correlated or uncorrelated. At the user’s option, the spectral densities due to all sources, assumed independent, may be combined by means of Eq. 1-172. The autocorrelation function is computed by the following approximation to Eq. 1-166.

Equation 1-175. which assumes that Sj (ω) varies linearly between ωi and ωi + 1 and also assumes that Sj (ω) = 0 for ω < ω1 and ω > ωN . The user specifies the sequence of values of τ. The rms value of the response, u¯ j , is evaluated as the square root of a trapezoidal approximation to the integral in Eq. 1-167, i.e.,

Equation 1-176. The mean frequency, N0 , is evaluated from Eq. 1-168, using a trapezoidal approximation to the curve for Sj (ω)

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Equation 1-177. with

Equation 1-178.

Equation 1-179.

Equation 1-180. The power spectral densities, Sj , are plotted versus frequency and the autocorrelation functions, Rj (τ), are plotted versus the time delay, τ, at the user’s request. Cross-correlation functions and cross-spectral densities between different output quantities are not calculated. The measured power spectral density function for atmospheric turbulence has been fitted with analytic functions by several authors. Two of the commonly used functions are those of Dryden and von Karman [see, e.g., Taylor (1965, pp. 200-202)]. They can both be expressed by the equation,

Equation 1-181. where: Sa (ω) wg ω L

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= = = =

power spectral density (units of 1/(Hz)) RMS gust velocity circular frequency scale of turbulence (length units)

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V

=

airplane velocity (velocity units)

The values of the parameters k and p are given in the following table: k

Dryden 1.0

von Karman 1.339

p

1/2

1/3

A special data entry (TABRNDG) is used to select this analytic form in NX Nastran. The user supplies wg , L, and V, and selects either the Dryden or the von Karman parameters.

1.8

Aeroelastic Design Sensitivities and Optimization

Design sensitivity analysis is used to compute the rate of change, or first derivative, of a particular response quantity with respect to a change in a given structural parameter, or design variable. In NX Nastran, these responses may be computed using a number of analysis disciplines: statics, normal modes, buckling, direct and modal frequency response, and modal transient response. Static aeroelastic and flutter responses are available.

Static Aeroelastic Sensitivity Static aeroelasticity can be considered a special case of standard static structural analysis with the equilibrium equation

Equation 1-182. In static aeroelasticity, the stiffness matrix contains inertial terms related to quasi-rigid body accelerations and aerodynamic terms from the structural deformations and from the extra degrees of freedom related to aerodynamic variables. The sensitivity of Eq. 1-182 is simply its partial derivative with respect to a design variable x

Equation 1-183. where Δ is used to denote partial differentiation with respect to the design variable, i.e., Δ = ∂/∂x. This sensitivity calculation can be performed in much the same way as the static sensitivity analysis described in the NX Nastran Design Sensitivity and Optimization User’s Guide . The details of the static aeroelastic sensitivity follow. To explain the sensitivity analysis, it is useful to start from the static aeroelastic equations in the g-set

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Equation 1-184. The A matrices, which contain the aerodynamics, are not actually formed in the g-set in NX Nastran but are shown here to initiate the following discussion. The mean-axis condition that is used in static aeroelasticity [see Eq. 1-82] imposes additional requirements on the sensitivity equation. This orthogonality condition is imposed in the a-set and can be written as

Equation 1-185. If Eq. 1-182 is reduced to the a-set and coupled with Eq. 1-183 the basic equation for static aeroelasticity is obtained [compare with Eq. 1-84].

Equation 1-186. The basic sensitivity equation for static aeroelasticity has a similar form

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Equation 1-187. where the PL terms represent the pseudoloads caused by modifications to the structure. For static aeroelasticity, the total perturbed load in the g-set can be developed based on Eq. 1-184.

Equation 1-188. The first term on the right-hand size is the sensitivity of the applied loads to the design variables. Gravity and thermal type loadings can be affected by the design variables. Note that aerodynamic terms are not included in the load sensitivity equation since these terms are invariant with respect to the design variables. The DML term of Eq. 1-187 results from the variation of the mean-axis constraint of Eq. 1-185. The D matrix of the latter equation in invariant with respect to the structural parameter changes so that the sensitivity is simply

Equation 1-189. This equation points to the requirement for a vector of the form

Equation 1-190. The reduction of the pseudoload to the a-set required for Eq. 1-187 follows standard reduction techniques. A key point is that the matrix on the left-hand side of Eq. 1-187 is the same as the left-hand matrix of Eq. 1-186 so that operations already performed on this matrix as part of the analysis do not have to be repeated for the sensitivity calculations.

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In applying Eq. 1-187, there are three types of responses for static aeroelasticity for which sensitivity analysis is supported and the sensitivity calculation varies slightly for each. The first type is for the responses already available from static sensitivity, e.g., displacements, forces, and stresses. The sensitivity of these responses is derived based on sensitivities of the elastic deformations. These sensitivities are recovered from the Δul and Δur terms of Eq. 1-187 while using the displacement and accelerations vectors that were computed during the recovery phase of the trim analysis. A simplification results from the consideration that the static aeroelastic solution is performed relative to SUPORTed degrees of freedom so that ur , and therefore Δur , are zero in the sensitivity equation. The second type of response is for the TRIM variables. The sensitivities of these displacements are obtained directly in Eq. 1-187 and as a prerequisite to the first type of response discussed above. The third type of response is for Stability Derivatives. These derivatives are independent of the trim deformations and therefore require significant special code to accommodate their sensitivity calculation. That is, pseudodisplacements (and accelerations) are required that represent the deflections (and accelerations) that would occur if the aerodynamic extra point associated with the stability derivative were deflected a unit value. The calculation also differs depending on whether the user-requested stability derivative is restrained or unrestrained. In the restrained case, the accelerations and the displacements of the SUPORTed degrees of freedom are zero, whereas the unrestrained case has a complete set of displacements and accelerations. Stated in equation form, the pseudoloads of Eq. 1-187 that are obtained for the unrestrained derivatives are computed based on accelerations and displacements that result from a solution of the equation

Equation 1-191. where the us superscript indicates that the ux variable and the response quantities are associated with unrestrained stability derivatives. The ux vector has all of its rows set to zero except for the row associated with the variable of interest, which has a value of unity. Once the accelerations in the r-set and the displacements in the a-set have been computed, they can be recovered to the g-set using standard recovery techniques. The unrestrained stability derivative calculation can be viewed as the solution of the third row of Eq. 1-191.

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Equation 1-192. where the KAZ notation implies DTKali + Kari . Eq. 1-192 represents a dimensional equation, with the nondimensionalization a function of the stability derivative of interest. The sensitivity of Eq. 1-192 can be performed on either of the two right-hand sides. The second equality has been chosen giving

Equation 1-193. where the sensitivities of the displacements are obtained from the solution of Eq. 1-187. The following derivation utilizes notation used in the SAERSENS subDMAP (see Aeroelastic Solution Sequences ) and also notation already introduced in Unrestrained Stability Derivatives. The Δux term of Eq. 1-187 is zero so that the equation becomes

Equation 1-194. where DPSALU is the l-partition of the DPSAAU matrix that is, in turn, made up of the relevant columns from DPL and

Equation 1-195.

Equation 1-196.

Equation 1-197.

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Equation 1-198.

Equation 1-199. The first row of Eq. 1-194 is solved for UDUL to give

Equation 1-200. where:

Equation 1-201. Substituting into the second row of Eq. 1-194 gives

Equation 1-202. where:

Equation 1-203. Then

Equation 1-204. where:

Equation 1-205. If the expression for UDUL from Eq. 1-200 is substituted into the third row of Eq. 1-194, then

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Equation 1-206. where:

Equation 1-207. Then the expression for UDUR from Eq. 1-204 can be substituted into Eq. 1-206 to give

Equation 1-208. where:

Equation 1-209. Given UDURDD from Eq. 1-208, UDUR is recovered from Eq. 1-202 and UDUL is recovered from Eq. 1-200. The dimensional stability derivative data are then calculated using

Equation 1-210. The DSTABU vector is passed in to module DSARSN to compute the stability derivative sensitivities. The restrained stability calculations are greatly simplified since ur , ux , and ür are all invariant in Eq. 1-187. Therefore, it is necessary only to solve for the sensitivity of ul . Eq. 1-187 then becomes

Equation 1-211. where DUALS is Δul and PLSTBL is a partition of the appropriate columns of the PA matrix. The dimensional stability derivative vector is then computed using

Equation 1-212.

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Flutter Sensitivity The flutter design capability has been developed based on two key considerations: 1. The PK-flutter method is the most appropriate for flutter design. 2. The damping values obtained from the PK-flutter analysis are the most appropriate design responses. The PK-flutter method performs the flutter analysis at user-specified velocities and is therefore ideal in a design task in that it allows the user to focus on a particular velocity range. The Kand KE-flutter methods, on the other hand, perform the flutter analysis at user-specified reduced frequencies with little control over the associated velocities. The selection of damping values on the design response conforms to the design specifications that are applied to the flutter behavior of air vehicles. Further, the tedious and error-prone process of determining the flutter velocity is avoided with this approach. Finally, by constraining the damping values over a range of frequencies, the possibility of “hump modes” is addressed. The eigenvalue problem for flutter is given in Eq. 1-128, as

Equation 1-213. The terms necessary for the constraint conditions and the derivation of the sensitivity are defined again here: these are the frequency ω, the transient decay rate coefficient γ (Note that the structural damping coefficient g = 2γ), and the complex eigenvalue p = ω(γ + i) = pR + pr . The design response that is used in NX Nastran to address flutter instabilities is the decay coefficient γ. The sensitivity calculation for Δγ is conceptually straightforward, but is algebraically intense. From the relations γ = pR /ω and pI = ω, the derivative is expressed in terms of the real and imaginary parts of the eigenvalue as

Equation 1-214. It remains to determine the sensitivities of the real and imaginary parts of the eigenvalue. The formulation to be used requires the definition of additional notation:

Equation 1-215. is the left eigenvector in the global set,

Equation 1-216.

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Fundamentals of Aeroelastic Analysis

is the right eigenvector in the global set, and

Equation 1-217. The Fgg matrix is shown in order to lead the discussion. Aerodynamic matrices are not available in the g-set and the discussion will show that they are not needed. The left eigenvector of Eq. 1-213 is required in the sensitivity analysis, which points out a requirement for an eigenvector extraction in the sensitivity phase that is in addition to the aeroelastic calculations currently performed. This is a relatively straightforward calculation since the eigenvalues of a matrix and its transpose are identical. Eq. 1-213 can be recast as:

Equation 1-218. Differentiating:

Equation 1-219. The term containing {Δuh }, the eigenvector sensitivity, is neglected in the following. This is equivalent to assuming that the normal mode eigenvectors adequately span the space over which the flutter responses vary. A rigorous simplification results from premultiplying Eq. 1-219 by vh T. The left-hand eigensolution of the flutter equation gives vg TFgg = 0 so that the third term in Eq. 1-219 becomes zero and the remaining equation is:

Equation 1-220. expanding:

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Equation 1-221. By writing the expressions in the g-set, the sensitivities of the mass, damping, and stiffness matrices are available. The computation of the sensitivities of the aerodynamic matrices is straightforward since the matrices are a function only of Mach number and reduced frequency. The Mach number is invariant in the calculation while the reduced frequency can vary. For the aerodynamic sensitivity calculation, it is convenient to introduce the following notation:

Equation 1-222. and

Equation 1-223. The sensitivities are then computed as:

Equation 1-224.

Equation 1-225. . To be rigorous, it should be noted that ΔRQhh = Re({vg }T[Gkg TΔQkk where Gkg ]{ug }). A similar result applies to ΔIQhh . The sensitivity of the aerodynamic matrix with respect to the reduced frequency can be computed analytically based on the spline fit used for matrix interpolation. This fit expresses the aerodynamics as the weighted sum of invariant matrices, with the weighting functions a cubic function of the reduced frequency. Eq. 1-221 can then be written as:

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Equation 1-226. Eq. 1-226 is a complex scalar equation with the complex Δp as the single unknown. The equation can be separated into its real and imaginary parts based on the following definitions:

Equation 1-227.

Equation 1-228.

Equation 1-229.

Equation 1-230.

Equation 1-231.

Equation 1-232.

Equation 1-233.

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Equation 1-234.

Equation 1-235.

Equation 1-236.

Equation 1-237. Then the real and imaginary parts of Eq. 1-226 give the following two equations.

Equation 1-238. where

Equation 1-239.

Equation 1-240.

Equation 1-241.

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Equation 1-242.

Equation 1-243.

Equation 1-244. where

and

All the matrices in these equations are real so that the scalar relationships are also real. The solution of Eq. 1-238 leads to ΔpR and ΔpI which are substituted into Eq. 1-214 to obtain the desired sensitivities. In the case where the eigenvalue is real, the definition of the decay coefficient is different from γ = pR /ω. In this case it is given by γ = pR /ln2 and the sensitivity becomes

Equation 1-245.

Optimization with Aeroelastic Constraints The foregoing aeroelastic sensitivities are calculated for each design variable, along with other structural sensitivities, in SOL 200 of NX Nastran. When the constraints are all specified, aeroelastic and otherwise, a specific objective (cost) function, e.g., weight, can be minimized via the optimization procedures in SOL 200 [see the NX Nastran Design Sensitivity and Optimization User’s Guide and Vanderplaats (1984)]. Guidelines for selection of aeroelastic design variables and constraints are presented in Aeroelastic Modeling. Example problems illustrating optimization for strength and aeroelastic characteristics are presented in Aeroelastic Design Sensitivities and Optimization.

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Chapter

2

Aeroelastic Modeling

•

Overview

•

Aerodynamic Modeling

•

Static Aeroelastic Analysis

•

Flutter Analysis

•

Dynamic Aeroelastic Response Analysis

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Chapter 2

2.1

Aeroelastic Modeling

Overview

The NX Nastran aeroelastic analysis and design capabilities are intended for the study of stability and response of aeroelastic systems. They are compatible with the general structural analysis and design capabilities of NX Nastran. This section deals primarily with aerodynamic data and the connection between structural and aerodynamic elements as well as the methods for stability and response analysis and design. Some discussion of both structural and aerodynamic modeling is also included with the example problems in Static Aeroelastic Analysis Sample Problems through Aeroelastic Design Sensitivities and Optimization. Aerodynamic Modeling deals with the aerodynamic data. The selection of a good aerodynamic model depends upon some knowledge of the theory (see Aerodynamic Theories ). Several aerodynamic theories are available; all assume small amplitude, quasi-steady or sinusoidal motions. Transient aerodynamic loads are obtained by Fourier methods. The Interpolation from Structural to Aerodynamic Models deals with the interpolation from structural to aerodynamic degrees of freedom, and the interpolation methods include both linear and surface splines. These methods are usually superior to high-order polynomials because they tend to give smoother interpolations since they are based upon the theory of uniform beams and plates of infinite extent. Static Aeroelastic Analysis describes static aeroelastic analysis. The trim solution is obtained only in quasi-steady equilibrium at subsonic or supersonic speeds. Static and dynamic aerodynamic stability derivatives, as well as aerodynamic and structural loads, are also obtained along with structural element stresses and deflections. Aeroelastic divergence speeds of restrained vehicles may be found from the K- or KE-methods of modal flutter analysis, or they may be obtained from a special purpose static aeroelastic option. Divergence and flutter speeds of any vehicle may always be determined by the PK-method of flutter analysis. Flutter Analysis describes modal flutter analysis by the three available methods: the American (K) method, a streamlined version of the American method (KE), and the British (PK) method. After a vibration analysis, the flutter frequencies and dampings are obtained as functions of the velocity, and the relative modal amplitudes are found, except in the KE-method. The physical displacements in the vibration modes are also available for output. Dynamic Aeroelastic Response Analysis gives instructions for modal dynamic aeroelastic response analyses. These include frequency response, transient response, and random response analyses. The excitation may consist of applied mechanical forces using any aerodynamic theory or gusts with the Doublet-Lattice and ZONA51 theories only. The response parameters can include loads, stresses, and displacements. Aeroelastic Design Sensitivity and Optimization provides guidance for adding static aeroelastic and flutter constraints to the NX Nastran optimization design procedure (SOL 200).

2.2

Aerodynamic Modeling

Aerodynamic elements are regions of lifting surfaces or bodies. Since the elements occur in regular streamwise arrays, the aerodynamic connection (CAEROi) Bulk Data entries have been designed to specify these arrays. The aerodynamic grid points associated with the elements in an array are generated internally in the program. Spline methods are used to interpolate aerodynamic grid point deflections to structural grid points. For every aerodynamic problem, pertinent basic flight and geometric parameters are specified on one of two Bulk Data entries: the AEROS entry is used in static aeroelastic analysis, and the AERO entry is used in dynamic analysis. A rectangular aerodynamic coordinate system must be identified. The flow is in the positive x-direction in this system and parallel to the plane of the

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aerodynamic elements. The use of symmetry (or antisymmetry) is available to analyze structures that have both stiffness and inertial symmetry, to simulate ground effects, and to simulate wind tunnel wall effects. Any consistent set of units can be used for the dimensional quantities. The types of elements available are shown in Table 2-1. Every CAEROi entry must reference an aerodynamic property (PAEROi) data entry that is used to list additional parameters. Tabulations of numbers or other defining parameters are sometimes required, depending on the selected aerodynamic method, and these are listed on AEFACT entries. These lists include division points (for unequal box sizes) and a variety of other parameter values. Table 2-1. NX Nastran Aerodynamic Elements Attribute

Aerodynamic Theory DoubletLifting Body ZONA51 Mach Box Lattice Panel (Interference) Surface Panel

Bulk Data

CAERO1

CAERO2

CAERO1

CAERO3

Entries

PAERO1

PAERO2

PAERO1

PAERO3

Mach Number

Subsonic

Subsonic

Two Planes

Two Planes

Symmetry Options y = 0

y=0

z=0

z=0

Interaction

Panels and Bodies in the Same Group

Interconnection to Box Centers Structure Displacement Components Used 3,5 at Connection Points

2,6 y-Bodies

Piston Theory

CAERO4 CAERO5

PAERO4 PAERO5 High Supersonic Supersonic All Supersonic One Plane One Plane None Required y=0 Panels in the Same Group

Slender Body Box Centers Centers 3,5 z-Bodies

Strip Theory

3,5

None

Boxes on One None None Surface User Strip1/4- Strip1/4Specified Chord Cord Locations 3,5

3,5 and 6 for Control

3,5 and 6 for Control

Doublet-Lattice and ZONA51 Panels The configuration is divided into planar trapezoidal panels (macro-elements), each with a constant dihedral and with sides parallel to the airstream direction. These panels are further subdivided into "boxes" (see Figure 2-1), which are similarly configured trapezoids. The following are guidelines that are not enforced by the program; the user is expected to ensure adherence to these rules. If a surface lies in (or nearly in) the wake of another surface, then its spanwise divisions should lie along the divisions of the upstream surface. The strips near the intersection of intersecting surfaces should have comparable widths. The aspect ratio of the boxes should approximate unity; less than three is acceptable in the subsonic case, and of order one is desirable in the supersonic case. The chord length of the boxes should be less than 0.08 times the velocity divided by the greatest frequency (in Hz) of interest, i.e., Δx < 0.08V/f (Note that this is a requirement for approximately 12 boxes per minimum wavelength; however, no less than four boxes per chord should be used.) Boxes should be concentrated near wing edges and hinge lines or any

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other place where downwash is discontinuous and pressures have large gradients. [Note that concentrating boxes near hinge lines is a requirement of Potential Theory (which neglects viscous effects); not increasing the concentration of boxes near hinge lines lowers the calculated control surface effectiveness and leads to closer agreement with experimental data (with the added benefit of reduced computational time).] The chord lengths of adjacent boxes in the streamwise direction should change gradually. A further discussion of the choice of models is found in Rodden, Harder, and Bellinger (1979).

Figure 2-1. An Aerodynamic Doublet-Lattice/ZONA51 Panel Subdivided into Boxes Aerodynamic panels are assigned to interference groups. All panels within a group have aerodynamic interaction. The purpose of the groups is to reduce the computational effort for aerodynamic matrices when it is known that aerodynamic interference is important within the group but otherwise is negligible or to allow the analyst to investigate the effects of aerodynamic interference. Each panel is described by a CAERO1 Bulk Data entry. A property entry PAERO1 may be used to identify associated interference bodies in the subsonic case; if there are no bodies in a subsonic case, or in a supersonic case, a dummy PAERO1 entry is still required. A body should be identified as a member of the group if the panel is within one diameter of the surface of the body. The box divisions along the span are determined either by specifying the number of equal boxes NSPAN, or by identifying (by LSPAN), the AEFACT data entry that contains a list of division points in terms of fractions of the span. A similar arrangement is used to specify divisions in the chordwise direction by choosing NCHORD or LCHORD. The locations of the two leading edge points are specified in the coordinate system (CP) defined by the user (including basic). The lengths of the sides (chords) are specified by the user, and they are in the airstream direction (i.e. parallel to the x-axis of the aerodynamic coordinate system specified on the AERO or AEROS Bulk Data entry). Every panel must be assigned to some interference group (IGID). If all panels interact, then IGID must be the same for all panels. Aerodynamic Grid Points

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There is an aerodynamic grid point with its associated degrees of freedom in plunge and pitch for each box within a given panel. These points are located at the center of each box and are automatically numbered and sequenced by the program. The lowest aerodynamic grid point number for a given panel is automatically assigned the same number as specified for the panel ID field on the CAERO1 entry starting with the box connected to point 1. The grid point numbers increase in increments of 1 (see the CAERO1 Bulk Data entry description) first in the chordwise direction and then spanwise over all boxes in the panel. The user must be aware of these internally generated grid points and ensure that their numbers are greater than any structural grid, scalar, and extra point IDs. The reason for this is that these aerodynamic points are used for output including displacements, plotting, matrix prints, etc. The local displacement coordinate system has component T1 in the flow direction and component T3 in the direction normal to the panel in the element coordinate system defined on the CAERO1 entry.

Slender and Interference Bodies In subsonic problems, bodies are idealized as "slender" and "interference" elements in combination. The primary purpose of the slender body elements is to account for the forces arising from the motion of the body, whereas the interference elements are used to account for the interference among all bodies and panels in the same group. This is done by providing a surface through which the boundary condition of no flow is imposed. Bodies are further classified as to the type of motion allowed. In the aerodynamic coordinate system, y and z are perpendicular to the flow. In general, bodies may move in both the y- and z-directions. Frequently, a body (e.g., a fuselage) lies on a plane of symmetry and only z- (or y-) motion is allowed. Thus, any model may contain z-bodies, zy-bodies, and y-bodies. One or two planes of symmetry or antisymmetry may be specified. Figure 2-2 and Figure 2-3 show an idealization with bodies and panels. This example is the one used to illustrate the Doublet-Lattice program in Giesing, Kalman, and Rodden (1972b, pp. 19-42). It has a fuselage, a wing, a pylon, and a nacelle.

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Figure 2-2. Illustration of Boxes and Slender Body Elements.N5KA Bomber Example with Three Panels, Ten Boxes, Two Bodies,Nine Slender Body Elements, and Seven Interference Elements

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Figure 2-3. Illustration of Interference Elements.N5KA Bomber Example with Three Panels, Ten Boxes, Two Bodies,Nine Slender Body Elements, and Seven Interference Elements The PAERO1 Bulk Data entry lists the IDs of all the bodies that are associated (i.e., interfere) with a given Doublet-Lattice panel (CAERO1 entry). The CAERO2 entry specifies the geometry and divisions for the slender body and interference elements. The PAERO2 entry provides orientation and cross-section data for the slender body and interference elements as well as the sampling data to account for the residual flow discussed later. The location of the body nose and the length in the flow direction are given. The slender body elements and interference elements are distinct quantities and must be specified separately. At least two slender body elements and one interference element are required for each body. The geometry is given in terms of the element division points, the associated width, and a single height-to-width ratio for the entire body length. The locations of the division points may be given in dimensionless units or, if the lengths are equal, only the number of elements need be specified. The body may be divided along its length unequally to characterize the lift distribution, noting that Slender Body Theory gives a lift proportional to the rate of change of cross-section area. Shorter elements should be chosen at the nose where the area is changing rapidly; longer elements can be used along cylindrical regions where the area is constant and intermediate length elements can be used in transition regions. The semiwidths of the slender body at interference element boundaries can be specified separately and are given in units of length. Usually the slender body semiwidth is taken as zero at the nose and is a function of x, while the interference body semiwidth is taken to be constant.

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The interference elements are intended for use only with panels and/or other bodies, while slender body elements can stand alone. Grid points are generated only for the slender body elements. The first grid point is assigned the ID of the body corresponding to the element at the nose and other grid points are incremented by one. The user must ensure that the IDs of these generated grid points are greater than any structural grid, scalar point, and extra point ID in the model, and greater than any other aerodynamic grid point ID. There are some requirements about bodies that have been imposed to simplify coding. All z-only bodies must have lower ID numbers than zy-bodies, which, in turn, must have lower ID numbers than y-only bodies. The total number of interference elements associated with a panel is limited to six. The user should be cautious about the use of associated interference bodies since they increase computational effort significantly. A brief review of the Method of Images (and its approximations) follows before the implementation of the method in NX Nastran is discussed. The interference elements provide the basis for the internal image system that cancels most of the effects of the trailing vortices from the lifting surfaces. Because of the two-dimensional basis for this approximation (Thompson’s Circle Theorem in Hydrodynamics), the body surface has been approximated by a constant elliptical cross-section cylinder called the interference tube, and it is this cylindrical tube that is divided into the interference elements. All panels that intersect a body must be attached to the interference tube. Image locations are computed from the semi-width of the interference tube for all lifting surfaces associated with the body. The image is only computed if it lies between the front of the first interference element and aft of the last interference element for the associated body. Shorter interference elements are placed in regions of substantial interference, e.g., near the wing-fuselage intersection; longer interference elements are placed in regions of less interference, and, of course, no elements are necessary where interference may be neglected. An image is only computed if it lies between the front of the first interference element and the rear of the last interference element for the associated body. The division of the interference tube into interference elements is independent of the division of the body into slender body elements; the longitudinal locations of their end points are independent, although they can be chosen to be the same for convenience. There is a residual flow "through" the body surface because the image system, being based on two-dimensional considerations, only partially cancels the flow through the body surface. It does not compensate for the effects of the bound vortices on the lifting surfaces or other bodies. Additional unknown "residual" doublets are located along the axis of the body, and, when determined, are added to the known doublet strengths of the slender body elements. The residual flow is calculated by "sampling" the vertical or side velocity components from the net effect of the surface, slender body, and image vortices or doublets. The sampling is performed at various angular positions around the periphery of the elliptical interference tube at the end points of the interference elements. Two sampling patterns can be specified: the first might be dense for a region of strong interference, the other might be sparse for a region of weak interference (or the roles of the two may be interchanged). The strengths of the "residual" doublets are then determined to cancel the net velocity. The calculation of the velocity field induced by the residual doublets requires knowledge of the geometry of the cross section of the slender body at the end points of the interference elements. However, experience shows that the residual flow is small compared to the slender body flow field so that the residual flow need not be represented accurately. This permits the further approximation of simply using the geometry of the constant cross-section interference tube in the calculation of the velocities induced by the residual doublets. The contents of the various fields of the CAERO2 and PAERO2 data entries may now be summarized. The CAERO2 entry defines the slender body element end points and the interference body end points, the coordinates of the body nose, and the body length. The PAERO2 entry defines the cross-sectional properties of the slender body and the interference

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tube: ORIENT specifies the direction(s) of motion; WIDTH is the half-width of the interference tube; AR is the body/tube aspect ratio (height/width); LRSB points to an AEFACT entry that lists the slender body half-widths at the end points of the slender body elements; LRIB points to an AEFACT entry that lists the slender body half-widths of the end points of the interference elements (note that because the residual flow is small, as discussed above, leaving LRIB blank results in the velocities induced by the residual doublets being based on the interference tube cross-section, specified by WIDTH and AR, without significant error, and is recommended); LTH1 and LTH2 point to AEFACT entries that list the angles θ1 and θ2 (in degrees), respectively, around the periphery of the elliptical interference tube at which the residual flow velocity components are sampled and averaged, the first being the dense (or sparse) sampling and the second being the sparse (or dense) sampling; finally, THIi and THNi list the first and last interference element (numbering beginning at one for each body) to use the θ1 -array. A discussion of two related problems follows: The requirement for a constant cross-section interference tube may require moving the stabilizer (or wing); see Giesing, Kalman, and Rodden (1972a, §2.5.8). NX Nastran can accommodate the requirements by specifying a stabilizer coordinate system, two sets of GRID points for the (same) stabilizer root and its fuselage connection, and MPCs constraining the motions of the two sets of GRIDs to be the same. In this way, the structure can be modeled faithfully although the aerodynamic model is only approximate. This is illustrated in FSW Airplaine with Bodies (Example HA144F). The idealization of a jet engine installation as a slender body results in a mass flow ratio through the engine of zero, since there is no flow through the body. Idealizing the engine as a ring-wing results in a mass flow ratio of unity, since all the flow goes through the tube. A typical mass flow ratio is 0.7, so a ring-wing representation is more appropriate.

Mach Box Method The supersonic lifting surface method for isolated surfaces that was originally included in the Aeroelastic Addition to MSC.Nastran [Rodden, Harder, and Bellinger (1979)] was a version of the Mach Box method developed for the NASA Space Shuttle. It has been superseded in the software by ZONA51 for multiple interfering surfaces, but has been retained to allow upward compatibility and for users who do not have the ZONA51 option (Aero II). Mach Box aerodynamics may be used to compute unsteady supersonic aerodynamic forces for a planar, isolated wing at supersonic speeds. The surface (see Figure 2-4) may have a leading and/or trailing edge crank. There may be one or two adjacent trailing edge control surfaces. The "inboard" edge (side 1-2 on the CAERO3 entry) must be the plane of structural, mass, and aerodynamic symmetry. The geometry of the planform is specified on the CAERO3 data entry. The two leading edge corners are located by the user. These, along with the flow direction, define the plane of the element. Up to 10 additional points are permitted to specify cranks and controls; these points are dimensional quantities using a coordinate system in the plane of the element and with its origin at point 1. The recommended minimum number of Mach boxes is 80. The recommended minimum number in the flow direction is seven. The number of boxes in the flow direction is entered on the PAERO3 entry. The number of spanwise boxes is determined within the program as outlined in Aerodynamic Theories since the total number of boxes depends on the Mach number, m, and will change for different Mach numbers. The number of Mach boxes in the flow direction may be computed as follows:

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where:

xmax = maximum chordwise dimension ymax = maximum spanwise dimension and Int( ) denotes the integer value of ( ), and NSB is the number of spanwise Mach boxes:

where NBOX0 = initial number of boxes selected In order to maintain aerodynamic matrices that are the same size for each m,k pair and that are generated at the same physical location, the user must select a set of aerodynamic grid points (also called control points in this case) to be used. On each aerodynamic surface there must be at least three noncolinear control points and usually more than three points are required to represent the modal deflections adequately. It is noteworthy that three noncolinear control points provide sufficient geometric information for rigid body motion of the aerodynamic surface. The CAERO3 entry selects the points by defining their geometric location on the wing. These aerodynamic grid points are located using the coordinate system shown in Figure 2-4. The T3 component of these points is normal to the plane of the element. Additional lists of at least three points are needed for each optional control surface that is included. These aerodynamic grid points are numbered starting with the ID field of the CAERO3 entry, which must be a larger ID number than any GRID, SPOINT, or EPOINT ID in the model. Interpolation from the Mach box centers to determine deflections and slopes at these designated control points is performed with surface spline routines within the program and requires no input from the user.

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Figure 2-4. Mach Box Surface The following restrictions apply to the Mach Box method: 1. The edge 1-2 is taken as a plane of structural, mass, and aerodynamic symmetry; either symmetric or antisymmetric motions can be considered. 2. Both leading edge and hinge line sweepback angles must be greater than or equal to zero. 3. If a leading edge crank is not present, then x5 , y5 do not have to be input. 4. If a trailing edge crank is not present, then x6 , y6 do not have to be input. 5. A trailing edge crank cannot be located on the trailing edge of a control surface. It must be located inboard of the inboard surface, outboard of the outboard surface, or exactly at the junction between the two control surfaces. 6. All control surface side edges must be parallel to the flow or swept inward. 7. Points 8, 10, and 12 are used with Points 7, 9, and 11, respectively, to define the control surface edge, and they must be distinct from Points 7, 9, and 11, but with one exception they do not have to lie on the wing trailing edge. The program will calculate new Points 8, 10, and 12 for the wing trailing edge. The exception is that Points 8, 10, or 12 must be located on the trailing edge if the trailing edge is cranked at the side edge of a control surface. 8. When only one control surface is present, it must be control surface one. 9. If the second control surface is not present, then x11 , y11 and x12 , y12 are not required as input. 10. If no control surfaces are present, then xi , yi (i = 7 through 12) are not required as input.

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Aeroelastic Modeling

11. No aerodynamic balance for the control surfaces has been included in the program. 12. The number of chordwise boxes used as input (NBOX) to the program should be carefully selected to provide at least 80 boxes on the wing but NBOX cannot exceed 50. Note that NBOX is the number of chordwise boxes between the most forward point and the most aft point on the lifting surface, as shown in Figure 2-5. 13. If the maximum number of allowable boxes (500 on the main surface, 200 on each control surface) is exceeded, the program will reduce the number of chordwise boxes one at a time until the number of boxes is under the allowable limit.

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Figure 2-5. Mach Box Surface Showing Mach Boxes and Diaphragm Region

Strip Theory Modified Strip Theory can be used for unsteady aerodynamic forces on a high aspect ratio lifting surface, although it is less accurate than the available lifting surface theories. Each strip may have two or three degrees of freedom. Plunge and pitch are always used, and rotation of an aerodynamically balanced control surface is optional. If a control surface is present, either a sealed or an open gap may be used. The planform (which may have several strips in one macro-element) is specified on a CAERO4 Bulk Data entry. A sample planform is shown in Figure 2-6. The user supplies the two leading edge corner locations and the edge chords as dimensional quantities. Edge chords are assumed to be in the flow direction. All additional geometry (box divisions, hinge locations, etc.) are given in dimensionless units. Multiple CAERO4 entries may be used if there are several surfaces or cranks. A grid point is assigned to each strip, and is assigned an ID starting with the CAERO4 entry ID and incrementing by one for each strip. The plunge (T3) and pitch (R2) degrees of freedom have the conventional definition. When a control surface is present, the R3 degree of freedom has a nonstandard definition, in the case of the relative control rotation. When interconnecting with the structure, the ordinary (surface or linear) splines can be used for T3 and R2, but a special method (see SPLINE3 data entry) is used for the relative control rotation.

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The parameters such as the lift curve slope or the lag function may be varied to account approximately for finite-span effects (three-dimensional flow) and Mach number by AEFACT Bulk Data entry selection from PAERO4. The AEFACT Bulk Data entry format used by Strip Theory is shown in the remarks on the PAERO4 Bulk Data entry. The user may request a Prandtl-Glauert (compressibility) and/or a sweep correction to the value of the lift curve slope. The lag function depends upon the local (i.e., using the chord of the strip) reduced frequency; for incompressible flow, it is the Theodorsen function C(k). An approximate form for this function is given by

Equation 2-1. in which 0 = 0, and may be selected for computing variations on the Theodorsen function that account for compressibility and finite span effects. The choice of parameters bn and 0 is left to the user to select values suitable for the requirement. Bisplinghoff, Ashley, and Halfman (1955, pp. 350, 394) give values for various Mach numbers and aspect ratios.

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Figure 2-6. Strip Theory Example Lifting Surface

Piston Theory Piston Theory in NX Nastran is a form of strip theory taken from Rodden, Farkas, Malcom, and Kliszewski (1962) who considered a rigid chord airfoil with an aerodynamically unbalanced rigid control surface. The aerodynamic forces are computed from third-order Piston Theory, which is

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valid for large Mach numbers m2 » 1, or sufficiently high reduced frequency m2k2 » 1. Although the latter condition may be met in subsonic flow, the primary application of Piston theory is in high supersonic flow. The coefficients of the point-pressure function (relating local pressure to local downwash) may be modified to agree with the Van Dyke theory for a rigid chord and to account for sweepback effects. The resulting strip parameters will depend upon the wing thickness distribution and spanwise variation of initial angle of attack, both of which must be supplied by the user. If a control surface is present, it is assumed to have no aerodynamic balance. Geometry specification and interconnection points follow the same rules as for Strip Theory. The additional information about angle of attack and thickness is input on AEFACT data entries that are referenced by the CAERO5 and PAERO5 data entries. The AEFACT data entry formats used by Piston Theory are shown in the Remarks on the PAERO5 data entry. If thickness integrals are input on AEFACT data entries, they must be calculated according to the thickness integral definitions on the CAERO5 data entry. The effect of wing thickness is to move the local aerodynamic center forward of the midchord; the effect of initial (trim) angle of attack is to move the local aerodynamic center aft toward the midchord. Conservative flutter speed predictions should result if the angle of attack is assumed to be zero. Chordwise flexibility (camber) can be approximated using multiple CAERO5 entries. Two examples (HA145G and HA145HA) illustrate this technique in Flutter Analysis Sample Problems.

Ordering of J- and K-Set Degrees of Freedom As detailed in Aerodynamic Data Input and Generation, the aerodynamic models used in NX Nastran generate two sets of data, namely, the j-set and the k-set. Typically, the user does not have a need to be able to determine which member of a matrix belongs to a particular aerodynamic element. The Case Control commands for pressures and forces (APRES and AEROF) are expressed in terms of the ID numbers input on the CAEROi entry. If, however, you need to manipulate matrices generated in the software (e.g., to generate running loads along a wing structure), the relationship between the matrix terms and the element locations is required. The order of both sets is driven by the ACPT data block (see Selected Aeroelastic Data Blocks ), each record of which corresponds to a CAEROi Bulk Data entry. For Strip Theory and Piston Theory, the k-set and j-set degrees of freedom are two aerodynamic points per strip (three per strip, if there is a control surface corresponding to: (1) plunge at the strip quarter chord, (2) rotation at the strip quarter chord, and (3) rotation of the control surface. Forces and moments are then (1) lift at the strip quarter chord, (2) pitching moment about the strip quarter chord, and (3) hinge moment about the control surface hinge line. For the Mach Box aerodynamic theory, the number of j- and k-set degrees of freedom are identical and are equal to the number of user-defined structural interpolation points. Points on the wing are given first, followed by points on the first (optional) controls surface, and then points on the second (optional) control surface. As detailed in Aerodynamic Data Input and Generation , the location of the j-set and k-set degrees of freedom differ for the Doublet-Lattice and ZONA51 aerodynamic theories. When the Doublet-Lattice method includes bodies, the number of degrees of freedom in the two sets differs as well. Figure 2-7 provides the algorithm for determining k-set degrees of freedom while Figure 2-8 provides the algorithm for determining j-set degrees of freedom.

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Figure 2-7. Order of k-Set Degrees of Freedom with Doublet-Latticewith Bodies and ZONA51

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Figure 2-8. Order of j-Set Degrees of Freedom with Doublet-Latticewith Bodies and ZONA51

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The Interpolation from Structural to Aerodynamic Models The interpolation from the structural to aerodynamic degrees of freedom is based upon the theory of splines (Figure 2-9). High aspect ratio wings, bodies, or other beam-like structures should use linear splines (SPLINE2). Low aspect ratio wings, where the structural grid points are distributed over an area, should use surface splines (SPLINE1). Several splines can be used to interpolate to the boxes on a panel or elements on a body; however, each aerodynamic box or element can be referenced by only one spline. Any box or body element not referenced by a spline will be “fixed” and have no motion, and forces on these boxes or elements will not be applied to the structure. A linear relationship (like an MPC) may be specified for any aerodynamic point using the SPLINE3 entry. This is particularly useful for control surface rotations. For all types of splines, the user must specify the structural degrees of freedom and the aerodynamic points involved. The given structural points can be specified by a list (SET1) or by specifying a volume in space and determining all the grid points in the volume (SET2). The degrees of freedom utilized at the grid points include only the normal displacements for surface splines. For linear splines, the normal displacement is always used and, by user option, torsional rotations and/or slopes may be included. Surface Splines The SPLINE1 data entry defines a surface spline. This can interpolate for any “rectangular” subarray of boxes on a panel. For example, one spline can be used for the inboard end of a panel and another for the outboard end. The interpolated aerodynamic degrees of freedom (k-set) are specified by naming the lowest and highest aerodynamic grid point numbers in the area to be splined. The two methods for specifying the structural grid points use either SET1 or SET2 data entries. A parameter DZ is used to allow smoothing of the spline fit. If DZ = 0 (the recommended value), the spline will pass through all deflected grid points. If DZ>0, then the spline (a plate) is attached to the deflected grid points via springs, which produce a smoother interpolation that does not necessarily pass through any of the points. The flexibility of the springs is proportional to DZ.

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Figure 2-9. Splines and Their Coordinate Systems Linear Splines

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The SPLINE2 data entry defines a linear spline. As can be seen from Figure 2-9, this is a generalization of a simple beam spline to allow for interpolation over an area. It corresponds to the frequently used assumption of the “elastic axis” in which the structure is assumed to twist about the axis such that the airfoil chord perpendicular to the axis behaves as if it were rigid. The portion of a panel to be interpolated and the set of structural points are determined in the similar manner as with SPLINE1. However, a coordinate system must also be supplied to determine the axis of the spline; a coordinate system with its y-axis collinear with the spline axis is required. Since the spline has torsion and bending flexibility, the user may specify the ratio DTOR of flexibilities for a wing as a representative value of EI/GJ; the default value for this ratio is 1.0. The attachment flexibilities, Dz , Dθx , and Dθy allow for smoothing, but usually all values are taken to be zero; when the attachment flexibilities are taken to be zero, the spline passes through all of the connected grid points and the value of the ratio DTOR has no effect. In the case where the structural model does not have one or both slopes defined, the convention DTHX = -1.0 and/or DTHY = -1.0 is used. When used with bodies, there is no torsion and the spline axis is along the body so that a user input coordinate system is not required. There are special cases with splines where attachment flexibility is either required or should not be used. The following special cases should be noted: 1. Two or more grid points, when projected onto the plane of the element (or the axis of a body), may have the same location. To avoid a singular interpolation matrix, a positive attachment flexibility must be used (or only one grid point selected at that location). 2. With linear splines, three deflections with the same spline y-coordinate overdetermine the interpolated deflections since the perpendicular arms are rigid. A positive DZ is needed to make the interpolation matrix nonsingular. 3. With linear splines, two slopes (or twists) at the same y-coordinate lead to a singular interpolation matrix. Use DTHX>0 (or DTHY>0) to allow interpolation. 4. For some modeling techniques, i.e., those which use only displacement degrees of freedom, the rotations of the structural model are constrained to zero to avoid matrix singularities. If a linear spline is used, the rotational constraints should not be enforced to these zero values. When used for panels, negative values of DTHX will disconnect the slope, and negative values of DTHY will disconnect the twist. For bodies, DTHY constrains the slopes since there is no twist degree of freedom for body interpolation.

2.3

Static Aeroelastic Analysis

Static aeroelastic analysis is intended to obtain both structural and aerodynamic data. The structural data of interest include loads, deflections, and stresses. The aerodynamic data include stability and control derivatives, trim conditions, and pressures and forces. The analysis presupposes a structural model (both stiffness and inertial data), an aerodynamic model, and the interconnection between the two. The requirements for static aeroelastic analysis beyond those for the structural and aerodynamic models are nominal. The Executive Control requires the statement SOL 144 to call for the Static Aeroelastic Response DMAP sequence, SOL 144. The stability derivatives are obtained as part of the solution process and are always printed if there are “SUPORT”ed degrees of freedom. All other derived quantities of interest must be requested in the Case Control: The subcase commands APRES = n and AEROF = n request the aerodynamic pressures and forces for a set of aerodynamic grid points defined by n. The flight conditions are specified for each SUBCASE on TRIM commands that involve TRIM Bulk Data entries.

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The AESTAT entries define the trim parameters, e.g., angle of attack α = ANGLEA, pitching velocity q = PITCH, normal acceleration = URDD3, and pitching acceleration = URDD5, in the longitudinal case. The trimming surfaces are specified on the AESURF entry; e.g., if the elevator deflection, δe = ELEV, is used to trim a symmetric flight condition, then both ELEV and the aerodynamic boxes in the aerodynamic model that lie on the elevator are identified on the AESURF entry. The reference geometry for the dimensionless stability derivatives and the symmetry condition for the maneuver are specified on the AEROS entry. Finally, the flight condition Mach number m and dynamic pressure q are specified on the TRIM entry along with all trim parameters that are known from the maneuvering condition; e.g., a longitudinal maneuver may specify q, , and and the trim solution would then determine α and δe . The AELINK entry can be used to specify known relationships among AESTAT and AESURF parameters. For example, inboard and outboard ailerons could be linked together in a schedule specified by flight control engineers. Experimental data can be included in the static aeroelastic analysis as discussed at the end of Aerodynamic Theories. Provision has been made for the premultiplying diagonal matrix [Wkk ]. to be input as DMI Provision has also been made for the experimental pressure coefficients with the name FA2J. The elements of WKK and FA2J are user-supplied data. An initial incidence, camber or twist distribution is introduced in Eq. 1-2 as {wj g} for inclusion in static aeroelastic analyses, and it can be input on DMI entries with the name W2GJ. The values must be derived by the user at the aerodynamic grid points of all aerodynamic boxes (at the box centerline midchord) and slender body elements (at the element midchord). No means of interpolation have been provided. Sample problems are presented in Static Aeroelastic Analysis Sample Problems to illustrate static aeroelastic trim solutions for symmetric, antisymmetric, and unsymmetric flight conditions.

Divergence Analysis The Static Aeroelastic Solution Sequence can also perform a divergence analysis. The analysis is invoked by a DIVERG command in Case Control which, in turn, invokes a DIVERG Bulk Data entry. The eigenanalysis of Eq. 1-101 is carried out using a complex eigensolver. A CMETHOD Case Control command invokes an EIGC Bulk Data entry that specifies the attributes for the eigenanalysis. The DIVERG Bulk Data entry allows the user to extract a desired number of divergence pressures (typically one, since the second and higher pressures are not of practical interest) for the Mach numbers given on the entry. Sample Case Control commands and Bulk Data entries for a divergence analysis are given in Listing 2-1. The CMETHOD request invokes a complex Lanczos eigenanalysis that asks for five roots to be extracted. The DIVERG Bulk Data entry indicates that the analysis is to be performed using incompressible aerodynamics (m = 0.0) and that five divergence roots are requested. Results from executing this input file are given at the end of Static Aeroelasticity in lieu of an example in Static Aeroelastic Analysis Sample Problems.

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TITLE = EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS S UBTI = DIVERGENCE ANALYSIS ECHO = BOTH SPC = 13 $ MPC = 1 $ CONTROL SURFACE RELATIVE MOTION SET 2 = 7 THRU 12 SET 3 = 11 DISP = 2 SPCF = 3 AEROF = ALL APRES = ALL DIVERG = 100 CMETHOD = 100 BEGIN BULK . . . DIVERG 100 5 0.0 EIGC 100 CLAN MAX 5 $ ENDDATA

Listing 2-1. Divergence Analysis Sample

2.4

Flutter Analysis

A flutter analysis determines the dynamic stability of an aeroelastic system. As noted in Introduction , three methods of analysis are available: the American (K) method, a restricted but more efficient American (KE) method, and the British (PK) method. The British method not only determines stability boundaries but provides approximate, but realistic, estimates of system damping at subcritical speeds that can be used to monitor flight flutter tests. The system dampings obtained from the K- and KE-methods is a mathematical quantity not easily related to the physical system damping. As with static aeroelastic analysis, flutter analysis presupposes a structural model, an aerodynamic model, and their interconnection by splines. The modal technique is used to reduce the number of degrees of freedom in the stability analysis. It should be appreciated by the user that the use of vibration modes for this purpose constitutes a series solution, and that a sufficient number of modes must be used to obtain convergence to the required accuracy. An aspect of the modal method is a transformation of the aerodynamic influence coefficients into modal coordinates. For computational efficiency, this transformation is carried out explicitly for only a few Mach numbers (m) and reduced frequencies (k). These generalized (modal) aerodynamic force coefficient matrices are then interpolated to any additional Mach numbers and reduced frequencies required by the flutter analysis. Matrix interpolation is an automatic feature of the program. The MKAERO1 and MKAERO2 Bulk Data entries allow the selection of parameters for the explicit calculations of the aerodynamic matrices. Features of the three flutter methods are shown in Table 2-2. Because of the iterative nature of the PK-method, the manner of convergence is frequently of interest to the user and can be seen by specifying the diagnostic, DIAG 39, in the Executive Control Section. The Case Control selects the flutter method on the FMETHOD command. It also selects the real eigenvalue method on a METHOD command for use in finding the vibration modes and frequencies for the modal flutter analysis. If the K-method of flutter analysis is to be used, a CMETHOD command is also required to specify the complex eigenvalue method. Plot requests may also be made in the Case Control for the frequency and damping versus velocity curves.

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Brief descriptions of the K-, KE-, and PK-flutter methods follow, before the requirements of the Bulk Data entries are summarized. Table 2-2. Flutter Analysis Methods Feature

K

K (complex) B (complex) M (complex) Aerodynamic Matrices M (complex) Structural Matrices

User Input Loops

Output

Method

Eigenvalue Method

Method KE

PK

K (complex)

K (real) B (real) M (complex) M (real) M (complex) K (real) B (real) ρ-Density ρ-Density ρ-Density m-Mach Number m-Mach Number m-Mach Number k-Reduced Frequency k-Reduced Frequency V-Velocity V-g Curve V-g Curve V-g Curve Complex Modes Complex Modes Displacements Displacements Deformed Plots Deformed Plots Compute Roots for User For Each ρ, m, V, Compute Roots for User Input ρ, m, k. Reorder Iterate on Each Root Input ρ, m, k Output so a “Curve” to Find Consistent Refers to a Mode. Results. Several Methods Complex Upper* Real Upper* Available, Selected Hessenberg Hessenberg by User via CMETHOD in Case Control

* No CMETHOD entry is used. All methods allow looping through three sets of parameters: density ratio ρ/ρref (ρref is given on an AERO data entry), Mach number m, and reduced frequency k. For example, if the user specifies two values of each, there will be eight analyses in the following order: LOOP (CURVE) 1 2 3 4 5 6 7 8

DENS 1 2 1 2 1 2 1 2

MACH 1 1 1 1 2 2 2 2

REFREQ orVELOCITY 1 1 2 2 I 1 2 2

The K-method of flutter analysis considers the aerodynamic loads as complex masses, and the flutter analysis becomes a vibration analysis using complex arithmetic to determine the

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frequencies and artificial dampings required to sustain the assumed harmonic motion. This is the reason the solution damping is not physical. When a B matrix is present, complex conjugate pairs of roots are no longer produced. NX Nastran uses the CEAD module to extract all requested roots but only selects roots with a positive imaginary part for the flutter summary output. alues for the parameters are listed on FLFACT Bulk Data entries. Usually, one or two of the parameters will have only a single value. Caution: If a large number of loops are specified, they may take an excessive time to execute. Multiple subcases can be specified to pinpoint particular regions for study while controlling CPU resources. The KE-method is similar to the K-method. By restricting the functionality, the KE-method is a more efficient K-method. The two major restrictions are that no damping (B) matrix is allowed and no eigenvector recovery is made. A complex stiffness matrix can be used to include the effects of structural damping. The KE-method therefore cannot consider control systems in which damping terms are usually essential, but it is a good method for producing a large number of points for the classical V-g curve of a system without automatic controls. The KE-method also sorts the data for plotting. A plot request for one curve gives all of the reduced frequencies for a mode whereas a similar request in the K-method gives all of the modes at one k value. Use of the alternative method for the specification of k (see the FLFACT Bulk Data entry) is designed to produce well-behaved V-g curves for the KE-method. The PK-method treats the aerodynamic matrices as real frequency dependent springs and dampers. A frequency is estimated, and the eigenvalues are found. From an eigenvalue, a new frequency is found. The convergence to a consistent root is rapid. Advantages of the method are that it permits control systems analysis and that the damping values obtained at subcritical flutter conditions appear to be more representative of the physical damping. Another advantage occurs when the stability at a specified velocity is required since many fewer eigenvalue analyses are needed to find the behavior at one velocity. The input data for the PK-method also allows looping, as in the K-method. The inner loop of the user data is on velocity, with Mach number and density on the outer loops. Thus, finding the effects of variations in one or both of the two parameters in one run is possible. The flight condition and remaining flutter control specifications are in the Bulk Data input. The AERO entry gives the basic aerodynamic data. The MKAERO1 or MKAERO2 entries specify the Mach number and reduced frequencies for which the generalized aerodynamic forces are computed explicitly. The FLUTTER entry selects the method of flutter analysis and refers to FLFACT entries for density ratios, Mach numbers, and reduced frequencies (K- and KE-methods) or velocities (PK-method). The IMETH field on the FLUTTER entry specifies the aerodynamic interpolation method. As discussed in Dynamic Aeroelastic Analysis , the K- and KE-methods of flutter analysis allow the user to select a linear spline that interpolates on reduced frequency for aerodynamic matrices at the Mach number closest to the required Mach number, or a surface spline that interpolates on both Mach number and reduced frequency. It is recommended that the linear spline be used in most cases. The PK-method only supports the linear spline method. The EIGR entry selects the real eigenvalue method to obtain the vibration modes and frequencies, and, if the K-method of flutter analysis is to be used, an EIGC entry selects the complex eigenvalue method to obtain the flutter roots and modes. The number of vibration modes computed is specified on the EIGR entry, but the number needed in the flutter analysis should be determined by a convergence study. The parameters LMODES or LFREQ and HFREQ can be used to select the number of vibration modes to be used in the flutter analysis and can be varied to determine the accuracy of convergence. The NVALUE field on the FLUTTER entry can be used to limit Flutter Summary output. Finally, the parameter PARAM,VREF may be used to scale the output velocity. This can be used to convert from consistent units (e.g., in/sec) to any units the user may desire (e.g., knots) as determined from Vout = V/(VREF). If physical output (grid point deflections or element forces, plots, etc.) is desired these data can be recovered by using a Case Control command; for example, the physical displacements can

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be obtained with the DISPLACEMENT case control command, DISP = ALL. A selected subset of the cases can be obtained by the OFREQUENCY command. The selection is based upon the imaginary part of the eigenvalue: velocity in the K- or KE-method, frequency in the PK-method. Example problems that demonstrate the different methods of flutter analysis with the various aerodynamic theories are presented in Flutter Analysis Sample Problems.

2.5

Dynamic Aeroelastic Response Analysis

The purpose of dynamic aeroelastic response analysis is to study the reactions of an aeroelastic system to prescribed loads and displacements, and to atmospheric gust fields. The effects of a control system can also be assessed if its equations (transfer functions) have been included in the (aeroservoelastic) model. As in flutter analysis, the modal method is employed to reduce the computational effort. SOL 146 can solve for frequency response, random response, and transient response problems in the presence of an airstream. Examples of response problems in which aerodynamic effects should usually not be neglected include high speed landing loads, in-flight store ejection loads, and loads and accelerations in a gust field. The analyses of frequency response, random response, and transient response without aerodynamic effects are discussed in the NX Nastran Basic Dynamic Analysis User’s Guide . That guide should be consulted as a reference for the frequency response and random response analyses considered here. However, it only discusses transient response analysis by direct integration methods, whereas Fourier transform methods are employed in SOL 146 for aeroelastic response because the unsteady aerodynamic loads are calculated only in the frequency domain. NX Nastran utilizes two techniques in solving for the response to an enforced displacement: •

large mass method

•

Lagrange multiplier method

Both methods are also discussed in the NX Nastran Basic Dynamic Analysis User’s Guide. The Lagrange multiplier method requires less judgment and is computationally efficient with a small number of modes, so it is utilized effectively in some dynamic aeroelastic response analyses; it is discussed in the NX Nastran Basic Dynamic Analysis User’s Guide. The implementation in the Bulk Data Input for the various selections in the Case Control Section is illustrated in the example problems of Dynamic Aeroelastic Response Analysis.

Frequency Response Analysis A frequency response analysis is an integral part of random response analysis and transient analysis, and it can be of interest in its own right to obtain transfer functions for designing control systems. If only the frequency response is desired, the user must specify the frequency content of the loading via the RLOADi Bulk Data entry. If the loading is more conveniently specified in the time domain, specifically, on the TLOADi Bulk Data entry, the solution sequence will lead to the transient response and the intermediate result of the frequency response will not be output; a DMAP ALTER is required to output the frequency response data in this case. The input data for a dynamic aeroelastic response analysis are similar to those required for a flutter analysis. The Executive Control Section will include the SOL 146 command for the Dynamic Aeroelastic Response sequence.

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Case Control Commands The Case Control is used to select constraints, methods, and output. The METHOD command is required since it selects a method to compute the structural modes and frequencies that provide the modal basis for the response analysis. Some form of modal damping should be requested [see discussion of alternative representations of damping in the NX Nastran Basic Dynamic Analysis User’s Guide since all structures have some damping which can affect the results significantly. The complex structural damping can be used here since it is consistent with the harmonic assumptions of a frequency response analysis. A frequency selection FREQ is required, and some form of excitation, either loading (e.g., direct loads, DLOAD) or enforced displacement, is also required. The GUST Case Control command specifies a gust field. Output that can be requested includes solution set displacements (amplitudes of modes and extra points), physical displacements (grid points and extra points), constraint forces and aerodynamic loading, element forces and stresses, and plots. The user specifies the flight conditions in the Bulk Data Input. The same velocity is specified on both the AERO and the GUST data entries if a gust response is desired, while Mach number and dynamic pressure are supplied on the Bulk Data entries PARAM,MACH and PARAM,Q. The FREQi entry defines the set of frequencies for which the frequency response is performed.

Random Response Analysis For a random response analysis, the loading is specified in the frequency domain on the DLOAD Case Control command in conjunction with RLOADi entries, or by an enforced motion. To proceed with the random response analysis subsequent to the frequency response analysis, it is only necessary to specify the power spectral density of the excitation. A prescribed random loading is obtained by using the RANDOM Case Control command in conjunction with the RANDPS and TABRND1 Bulk Data entries that specify the input excitation power spectrum. For gust loading, the GUST Bulk Data entry specifies either the von Karman or Dryden spectrum on the TABRNDG Bulk Data entry or a tabulated power spectrum that is input via the TABRND1 Bulk Data entry. The output power spectral density is requested by the XYOUT plot commands in Case Control. The root mean square values of each selected output response and its expected frequency, N0 , are automatically printed when output power spectra are requested.

Transient Response Analysis For transient response analysis under a prescribed loading condition, the loading is specified on the DLOAD Case Control command in conjunction with TLOADi entries or by an enforced displacement. To proceed with the transient response analysis subsequent to the frequency response analysis, it is only necessary to specify the time history of the excitation. For a prescribed excitation, its time history is specified on a TLOADi entry. The output times are specified by the TSTEP Case Control command and Bulk Data entry. For transient response to a gust, the GUST Case Control command and Bulk Data entry augment the DLOAD Case Control command. A dummy DLOAD request is necessary to force the transient solution. The GUST Bulk Data entry now refers to a TLOADi entry that gives the gust profile. The Bulk Data entry PARAM,GUSTAERO,-1 is necessary to generate the airloads required for the gust load calculation. On a restart; the default value (+1) is recommended if no new gust loads are to be computed. It Is also recommended that the gust aerodynamics be calculated at the same time the aerodynamics for motion are calculated, i.e., include PARAM,GUSTAERO,-1 in the initial flutter solution (SOL 145), so that all aerodynamics will be available on the database for a restart in SOL 146. Generating gust aerodynamics along with motion aerodynamics does not add significant cost, but generating them subsequently in a restart requires repeating all of the aerodynamic calculations.

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Aeroelastic Design Sensitivity and Optimization The NX Nastran Design Sensitivity and Optimization User’s Guide contains a comprehensive description of the NX Nastran design sensitivity and optimization capability. This section contains supplementary information on the aeroelastic aspects of this capability and is divided into subsections on analysis, response evaluation, sensitivity, and optimization. Multidisciplinary Analysis For an optimization procedure to be of maximum benefit, it must be able to simultaneously take into account of all the conditions that impact the design. For this reason, the design sensitivity and optimization capability in NX Nastran is based on a multidisciplinary analysis capability that includes statics, normal modes, buckling, direct and modal frequency, modal transient, static aeroelastic, and flutter analyses. The static aeroelastic and flutter analysis capabilities present in the multidisciplinary analysis and design solution sequence (SOL 200) contain the full capabilities of the static aeroelastic (SOL 144) and flutter (SOL 145) solution sequences. It is necessary in SOL 200 to designate the type of analysis being performed for each subcase using the ANALYSIS Case Control command. ANALYSIS = SAERO is used for static aeroelasticity, while ANALYSIS = FLUTTER is used for flutter analysis. Response Evaluation For a sensitivity value to be computed, the user must designate it on a DRESP1 entry and either constrain it on a DCONSTR entry or identify it as the design objective using the DESOBJ Case Control command. Further, the DCONSTR set must be selected by either a DESSUB or a DESGLB Case Control command. For static aeroelasticity, the DRESP1 entry can be used to invoke standard static analysis responses, specifically, RTYPE = DISP, STRAIN, STRESS, FORCE, CSTRAIN, CSTRESS, and/or CFAILURE, as well as two responses, RTYPE = STABDER and/or TRIM, that are unique to static aeroelasticity. The STABDER response requests a stability derivative response and therefore selects one of the components of an AESTAT or AESURF aerodynamic extra point. The selected response type can correspond to a restrained or unrestrained derivative (see Static Aeroelasticity ), based on the value of the ATTB field. The utility of this request is that it is possible to determine how a key aeroelastic parameter, such as lift curve slope,CLα , varies when a structural change is made. More significant perhaps, it is possible to include design requirements on these stability derivatives in an NX Nastran design optimization study. The TRIM response on the DRESP1 entry requests a particular aerodynamic extra point by referencing an AESTAT or AESURF entry ID. The associated response is the magnitude of the aerodynamic extra point for the maneuver condition defined for the subcase. It is to be expected that the sensitivity of this response to a particular structural parameter is small. The response can have utility in limiting the range over which an aerodynamic value can vary during an optimization task; e.g., by limiting an elevator rotation to be less than 20 degrees, therefore, unrealistic designs can be precluded. A final DRESP1 response type related to aeroelasticity is for flutter (RTYPE = FLUTTER). The entry selects damping values from an aerodynamic flutter analysis as response quantities. The ATTi (i = 1,2,3 and 4) fields of this entry allow for a precise selection of the damping values from the available responses; i.e., ATT1 specifies a SET1 entry that selects the mode set, ATT2 specifies an FLFACT entry that selects the set of densities, ATT3 specifies an FLFACT entry that selects a set of Mach numbers, and ATT4 specifies an FLFACT entry that specifies a list of velocities. The requested data must exist from the analysis at precisely the Mach number, density, and velocity triplets specified by the FLFACT data. The effective use of this capability requires knowledge of the flutter characteristics of the vehicle so that the subset of the analysis results that are selected for design are both reasonable and comprehensive. For example, it

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would make little sense to try and alter the structure to modify undesirable damping values that result from the rigid body response of the vehicle. Similarly, most damping values are noncritical and can be safely excluded from the design task. NX Nastran can also construct synthetic responses that can be a function of DRESP1 response values, design variables values, user-defined constants, and grid locations. This is done using a combination of the DRESP2 entry to define the quantities that contribute to the synthetic response and a DEQATN entry that provides the equation that defines the synthetic response. A particular aeroelastic application of this capability is the construction of a response that predicts the roll performance of the vehicle as a function of the ratio of two stability derivatives:

Obtaining adequate roll performance is often a design driver for air-combat vehicles and typically entails enhancing the torsional stiffness of a wing. By use of the above relationship, this requirement can be incorporated into an NX Nastran design task. An example of this is given in Aeroelastic Optimization of FSW Airplane (Example HA200A). Sensitivity Analysis The specification of response quantities as described in the preceding subsection is a means towards the end of obtaining information for the structural design task. The first type of information that is available is sensitivity results wherein the rate of change of a particular response quantity rj , with respect to a change in a design variable xj , is produced:

Defining the Design Variables in the NX Nastran Design Sensitivity and Optimization User’s Guide (Moore, Version 68) contains a detailed description of design sensitivity analysis while in this Aeroelastic Design Sensitivities and Optimization provides a description of the calculations required to provide these sensitivities for aeroelastic responses. This section provides guidelines useful in obtaining desired sensitivity information. The user selects sensitivity analysis by setting PARAM,OPTEXIT equal to 4. An example of the output obtained with this option is given in Aeroelastic Optimization of FSW Airplane (Example HA200A). The NX Nastran implementation of design sensitivity analysis requires that the responses specified on DRESP1 and DRESP2 entries must be “constrained” in order for design sensitivity to occur. Further, the constrained responses have to pass through screening criteria that are applied in NX Nastran in order to limit the number of responses that are used in a design sensitivity and/or optimization task. The constraint specification begins with a DESSUB Case Control command that identifies the constraint set that is to be applied to a particular subcase. The command invokes DCONADD and/or DCONSTR Bulk Data entries, where the optional DCONADD entry is used to collect DCONSTR sets applicable in the subcase and the DCONSTR entry selects the DRESPi entries and specifies lower and upper limits on the response value. The screening procedure selects the constraints that are greater than a threshold value with a further limitation that only a limited number of responses of a given type will be retained (see the Defining the Analysis Disciplines in the NX Nastran Design Sensitivity and Optimization User’s Guide). You can force a response to be retained by using the DSCREEN Bulk Data entry

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to reduce the threshold value and/or increase the number of retained responses. For sensitivity analysis, a trick that can be used to force the retention of a response is to specify identical upper and lower limits on the DCONSTR entry associated with the response. Optimization Once you have specified the design variables, a design objective, and design constraints, you can use NX Nastran to determine the design that provides the minimum (or maximum) value of the objective while satisfying the imposed constraints. This is a powerful tool for the aeroelastician in that it provides a systematic means of finding an improved design. An optimization task exploits any deficiencies in the analysis in a way that helps it achieve its goals. Again, see the NX Nastran Design Sensitivity and Optimization User’s Guide for guidelines on performing optimization tasks, including means of gaining insight into the performance of the optimizer. One user guideline that is relevant here is that the use of 0.0 as a limiting value on the DCONSTR entry should be avoided, if possible. NX Nastran uses a normalized value for the constraint that entails dividing the response value by the constraint limit. Specifying a limit of 0.0 then produces a division-by-zero problem that NX Nastran avoids by substituting a small number for the limit. In the context of aeroelasticity, the user would be inclined to apply an upper bound of 0.0 to a DRESP1 entry that has an RTYPE of FLUTTER. This would ensure a negative damping level. It is recommended that a DRESP2 entry be used to offset the flutter response from zero and also to scale the response so that the constraint varies over a wider range than the unscaled response. The DRESP2 response is of the form:

where γ is the flutter response, OFFSET is the offset value (typically 0.3) and GFACT is the scaling factor (typically 0.1). The DCONSTR entry would then impose an upper bound limit of -OFFSET/GFACT on the DRESP2 response, and this would be equivalent to restricting the flutter damping value to be negative.

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Chapter

3 Input Files for Aeroelastic Problems

•

Overview

•

Executive Control Section

•

Case Control Section

•

Bulk Data Section

•

Restarts

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Chapter 3

3.1

Input Files for Aeroelastic Problems

Overview

Before you can perform an aeroelastic analysis, you must have an input file for the finite element structural model that satisfies the descriptions in the NX Nastran Quick Reference Guide regarding the Executive Control statements, the Case Control commands, and the Bulk Data entries. The descriptions of the input files in this chapter contains only the information related to the features that must be included in the input file to obtain a static aeroelastic, flutter, or dynamic aeroelastic response analysis, or aeroelastic design sensitivity and optimization.

3.2

Executive Control Section

A user typically runs NX Nastran by invoking one of the standard solution sequences. These sequences are a collection of DMAP statements that drive the analysis and the aeroelastic sequences are described in Aeroelastic Solution Sequences. The aeroelastic analysis and design solution sequences are: SOLs 21 or 144 SOLs 75 or 145 SOLs 76 or 146 SOL 200

STATIC AEROELASTIC RESPONSE AERODYNAMIC FLUTTER DYNAMIC AEROELASTIC RESPONSE SENSITIVITY AND OPTIMIZATION

SOLs 21, 75, and 76 are referred to as “unstructured superelement solution sequences” since the associated DMAP is a single list of statements with no subDMAPs. These solution sequences are now obsolete and are not discussed further here. Solution 200, in addition to performing sensitivity and optimization, is also a multidisciplinary analysis procedure. This solution sequence contains the analysis capabilities of SOLs 101 (Statics), 103 (Normal Modes), 105 (Buckling), 108 (Direct Frequency Response), 111 (Modal Frequency Response), 112 (Modal Transient Response), 144 (Static Aeroelasticity), and 145 (Aerodynamic Flutter). This makes it possible to perform static aeroelastic and flutter analyses in a single run.

3.3

Case Control Section

The Case Control Section is described in Case Control Commands in the NX Nastran Quick Reference Guide. Fifteen Case Control data selection commands are available for aeroelastic analyses and design and various outputs. This section provides a brief description of each of the commands.

Static Aeroelastic Trim Variable Selection TRIM

Selects a TRIM Bulk Data entry in static aeroelastic response.

Divergence Solution Selection DIVERG

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Selects the number of eigenvalues and Mach numbers for the aeroelastic divergence analysis.

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Input Files for Aeroelastic Problems

Structural Damping and Transfer Function Selection SDAMPING

Selects a table that defines damping as a function of frequency.

TFL

Selects the transfer function set that represents a servomechanism.

Aeroelastic Flutter Solution Selection FMETHOD

Selects the method to be used in aerodynamic flutter analysis.

CMETHOD

Selects a complex eigenvalue method for flutter analysis.

METHOD

Selects a real eigenvalue method for vibration analysis.

Dynamic Aeroelastic Load Selection DLOAD

Selects the dynamic load to be applied in a transient or frequency response problem.

FREQUENCY

Selects the set of frequencies to be solved in frequency response problems.

GUST

Selects the gust load in aeroelastic response analysis.

RANDOM

Selects the RANDPS and RANDTi entries to be used in random analysis.

TSTEP

Selects integration and output time steps for transient problems.

Output Control AEROF

Requests the aerodynamic loads on the aerodynamic control points.

APRESSURE

Requests the aerodynamic pressures in static aeroelastic response.

SET

Lists identification numbers for output requests, or lists frequencies for which output will be printed in frequency response analysis.

3.4

Bulk Data Section

The Bulk Data entries required in the analysis of finite element models are described in the Bulk Data Entries in the NX Nastran Quick Reference Guide. This section provides a brief comment on those entries that are unique to NX Nastran’s aeroelastic capability. Table 3-1presents a list of these entries and indicates which ones are required (R) and available (A) in each of the solution sequences. SOL 146 typically requires significant dynamic analysis input [see the NX Nastran Basic Dynamic Analysis User’s Guide] and SOL 200 typically contains significant design model input [see the NX Nastran Design Sensitivity and Optimization User’s Guide that is not included here. Table 3-1. Bulk Data Entries for Aeroelasticity Bulk Data Entry AEFACT AELINK

Solution 144 A A

145 A

146 A

200 A A

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Table 3-1. Bulk Data Entries for Aeroelasticity Bulk Data Entry AELIST AERO AEROS AESTAT AESURF CAERO1 CAERO2 CAERO3 CAERO4 CAERO5 DIVERG FLFACT FLUTTER GUST MKAERO1 MKAERO2 PAERO1 PAERO2 PAERO3 PAERO4 PAERO5 PARAM SET1 SET2 SPLINE1 SPLINE2 SPLINE3 TABRNDG TRIM

Solution 144 A R A A A A(1)* A(2) A(2) A(2) A

145

146

R

R

A A A A A

A A A(3) A(3) A(3)

R R

A A(1) A(2) A(2) A(2) A A A A A A

R(4) R(4) A A A A A A A A A A A

A R(4) R(4) A A A(3) A(3) A(3) R(5) A A A A A A

A

200 A A A A A A A A A A A A A A A A A A A A A A A A A A A A

* Parenthetical numbers refer to the notes at the end of the table.

Notes: 1. CAERO2 and PAERO2 entries provides slender body aerodynamics and are only available for subsonic analyses. 2. In SOL 144, CAEROi and PAEROi for i = 3, 4, or 5 are not available for TRIM analysis; they can be used for DIVERGENCE analysis.

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3. In SOL 146, CAEROi and PAEROi for i = 3, 4, or 5 are not available for GUST analysis but can be used for dynamic aeroelastic analysis with nonaerodynamic loading (e.g., store ejection or landing loads). 4. At least one MKAERO1 or MKAERO2 entry must be present for flutter and dynamic aeroelastic analyses. 5. For dynamic aeroelastic analysis, the Q parameter is required. The descriptions of the Bulk Data entries found in the NX Nastran Quick Reference Guide are intended to be comprehensive so that extended descriptions are not presented here. Instead, brief comments are given which highlight features of the entry, particularly those that have been troublesome to users. AEFACT

Specifies lists of real numbers for the aerodynamic model required by the CAEROi and PAEROi entries. When the data refer to spanwise, chordwise, or bodywise division cuts, the data are in fractions of the span, chord, and body length, respectively.

AELINK

Links aerodynamic extra points.

AELIST

Defines aerodynamic elements associated with a control surface in static aeroelasticity.

AERO

Aerodynamic parameter for unsteady aerodynamics. Note that aerodynamic densities must be in consistent units; i.e., PARAM WTMASS does not apply to RHOREF.

AEROS

Aerodynamic parameters for steady aerodynamics. The reference area (REFS) is input for half the vehicle when a half-span model is used. The reference span (REFB) is always the full vehicle span.

AESTAT

Rigid body aerodynamic extra points. A number of prespecified labels are provided to invoke standard rigid body motions such as angle of attack or roll. If other labels are used, the burden is on the user to provide the corresponding forces.

AESURF

Aerodynamic control surface extra points.

CAERO1

Defines a wing panel for Doublet-Lattice and/or ZONA51. The leading edge locations are input in the CP coordinate system of the entry while the edge chords are in the aerodynamic coordinate system specified by the AERO or AEROS entry. This convention also applies to the other CAEROi entries.

CAERO2

Body data for Doublet-Lattice aerodynamics.

CAERO3

Panel data for Mach Box aerodynamics.

CAERO4

Panel data for Strip Theory aerodynamics.

CAERO5

Panel data for Piston Theory aerodynamics. Sweep corrections (NTHRY = 2) cannot be used when the leading edge is subsonic sec ≥M

DIVERG

Specifies static aeroelastic divergence analysis.

FLFACT

Specification of real number required in a flutter analysis or a flutter sensitivity analysis. The alternate form is useful for obtaining a good distribution of k-values for V-G flutter plots when using the KE-flutter method.

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FLUTTER

Specifies flutter analysis. IMETH = S permits interpolation of the computed aerodynamic data on both Mach number and reduced frequency when employing the K- or KE-methods of flutter analysis.

GUST

Specification of vertical gust parameters. The value of V on this entry must be identical with the VELOCITY input on the AERO entry.

MKAERO1

Specification of Mach number and reduced frequencies for aerodynamic matrix calculations.

MKAERO2

Alternate specifications of Mach numbers and reduced frequencies for aerodynamic matrix calculations.

PAERO1

Defines bodies associated with CAERO1 entries. A PAERO1 entry is required even when there are no bodies. Note that panels and bodies can be in the same interference group (IGID on the CAERO1 and CAERO2 entries) but not be associated. Associated bodies must be in the same interference group as the referencing panel.

PAERO2

Defines body properties. See Slender and Interference Bodies for an extensive discussion of this entry.

PAERO3

Additional specifications for Mach Box aerodynamics.

PAERO4

Additional specification for Strip Theory aerodynamics.

PARAM

This entry, which may also appear in the Case Control Section, is used to provide scalar values used in performing solutions. Parameters in theNX Nastran Quick Reference Guide has a comprehensive discussion of all the NX Nastran PARAMs.

SET1

Selects grids to be used in the splining of aerodynamics. This entry is also used to select flutter modes to be used in flutter sensitivity and optimization.

SET2

An alternative specification of the grids to be used in the splining of aerodynamics.

SPLINE1

Specification of the two-dimensional spline.

SPLINE2

Specification of the one-dimensional spline.

SPLINE3

Alternative specification of the splining between structural and aerodynamic grids. Although this is provided primarily for control surfaces, it has general applicability.

TABRNDG

Provides the simplified specification of atmospheric gust power spectral densities. The TABRND1 entry is available for the specification of spectra not supported by this entry.

TRIM

Specifies a trim flight condition.

3.5

Restarts

A powerful feature of NX Nastran is its ability to use previously computed results in a subsequent analysis. The “restart” capability is briefly described in Solution Sequences in the NX Nastran User’s Guide. This subsection first lists common scenarios in an aeroelastic analysis that would benefit from the restart capability and then provides input data, with examples, required to perform the restart.

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When to Use Restarts Restarts are of prime benefit when a given analysis requires significant resources in terms of CPU time, turnaround time, or computer disk space, and where the user expects to make a number of changes to the original analysis. In these situations it is desirable to make a primary run in a “cold” start and save the database for access in subsequent runs. In aeroelastic analyses, a significant portion of the resources are consumed in generating the aerodynamic matrices for the aerodynamic model. For this reason, restarts are recommended for the following aeroelastic operations: 1. Investigate changes in the aerodynamics model while leaving the structural model unchanged. 2. Investigate changes in the structural model while leaving the aerodynamic model unchanged. 3. Perform additional analyses using the same aerodynamic and structural models, such as: Change the velocities at which a PK-flutter analysis is performed. Change the dynamic pressure in an aeroelastic gust analysis. Perform a trim analysis at a new flight condition. 4. Add an m,k pair to refine the aerodynamic interpolation. 5. Continue a run after it has exceeded its time limit in the cold start. 6. Feed information from another solution sequence, such as Nonlinear Heat Transfer (SOL 106) into an aeroelastic solution.

When Not to Use Restarts With the performance available in present-day computers, resource limits are much less significant than they were in the recent past. For this reason, even the minor inconveniences caused by working with a database are often not justified, and the use of restarts is restricted to large analysis models acting in a “production” environment. Also, in the early stages of model development, it is likely that numerous changes will be made that will make restarting of minimal benefit.

Examples In order to perform a restart run, the cold start run has to have been performed and the database saved. This is done by submitting the cold start using the SCR = NO qualifier; e.g., nast68 ha44a scr=no

The examples presented below assume this cold start has been made. For the restart run, the database has to be invoked with a command of the form: nast68 ha144a_rst scr=no dbs=ha144a

Modifying the Aerodynamic Model Suppose it is desired to determine the effect of increasing the aspect ratio of the HA144A wing (see FSW Airplane in Level Flight (Example HA144A) from 4 to 5. Furthermore, suppose that there is no interest in the low dynamic pressure trim results (subcase) described in Example HA144A. This requires changes in the AEROS entry and the CAERO1 entry that models the

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wing, with the remaining Bulk Data entries unchanged. The input data file for the restart is shown in Listing 3-1. The restart simply entails the addition of a RESTART request in the File Management Section, the removal of the first subcase, and the replacement of the changed Bulk Data entries. The / entry removes entries on restart based on the sorted Bulk Data echo of the cold start. Note that for restart purposes the aerodynamic model is considered modified if any CAEROi, PAEROi, AEFACT, AERO, or AEROS Bulk Data entry is changed. If coordinate systems called out by these entries change, the modification is not detected, and a restart should not be used. RESTART VERSION=1,KEEP ID NXN, HA144A_RST TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO CEND TITLE = EXAMPLE HA144A: 30 DEG FWD SWEPT WING WITH CANARD SUBTI = SYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO LABEL = ASPECT RATIO INCREASE RELATIVE TO BASELINE ECHO = BOTH SPC = 1 $ SYMMETRIC CONSTRAINTS DISP = ALL $ PRINT ALL DISPLACEMENTS STRESS = ALL $ PRINT ALL STRESSES FORCE = ALL $ PRINT ALL FORCES AEROF = ALL $ PRINT ALL AERODYNAMIC FORCES APRES = ALL $ PRINT ALL AERODYNAMIC PRESSURES SUBCASE 2 TRIM = 2 $ 1 G LEVEL FLIGHT OUTPUT(PLOT) PLOTTER = NASTRAN SET 1 = ALL FIND SCALE, ORIGIN 1,SET 1 PLOT SET 1 PLOT STATIC DEFORMATION 0, ORIGIN 1, SET 1, OUTLINE BEGIN BULK / 2 AEROS 1 100 10.0 50.0 250. 1 / 10 11 1 +CAW CAERO1 1100 1000 8 4 +CAW 25. 0. 0. 10. 10.56624+25. 0. 10. ENDDATA

Listing 3-1. Restart Input File for Modifying the Aerodynamic Model of Example HA144A

Modifying the Structural Model Suppose that in the sweptback wing the flutter analysis of Subsonic Flutter Analysis of the 15-Degree Sweptback Wing by the KE-Method (Example HA145E) the effects of increasing the thickness of the plate from 0.041 to 0.05 were also to be investigated. This thickness value is input on the PSHELL entry as well as the leading and trailing edge CQUAD4 entries. Furthermore, no V-G plots are required in this reanalysis. Listing 3-2 shows the required restart input data file.

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Input Files for Aeroelastic Problems

RESTART VERSION=1,KEEP ID NXN, HA145E_RST TIME 5 $ SOL 145 $ FLUTTER ANALYSIS CEND TITLE = EXAMPLE HA145E: HALF SPAN 15-DEG SWEPT UNTAPERED WING SUBT = KE-METHOD FLUTTER ANALYSIS, DOUBLET-LATTICE AERO LABEL = 0.050 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES SEALL = ALL ECHO = BOTH SPC = 1 $ WING ROOT DEFLECTIONS AND PLATE IN-PLANE ROTATIONS FIXED SDAMP = 2000 METHOD = 10 $ MODIFIED GIVENS METHOD OF REAL EIGENVALUE EXTRACTION FMETHOD = 30 $ KE-FLUTTER METHOD SVEC = ALL $ PRINT VIBRATION MODES BEGIN BULK / 107 PSHELL 1 1 .050 1 1 / 8 21 CQUAD4 1 1 1 2 10 9 +M00000 +M00000 0.0 0.0 .050 .050 CQUAD4 2 1 2 3 11 10 +M00001 +M00001 0.0 0.0 .050 .050 CQUAD4 3 1 3 4 12 11 +M00002 +M00002 0.0 0.0 .050 .050 CQUAD4 4 1 4 5 13 12 +M00003 +M00003 0.0 0.0 .050 .050 CQUAD4 5 1 5 6 14 13 +M00004 +M00004 0.0 0.0 .050 .050 CQUAD4 6 1 6 7 15 14 +M00005 +M00005 0.0 0.0 .050 .050 CQUAD4 7 1 7 8 16 15 +M00006 +M00006 0.0 0.0 .050 .050 / 36 49 CQUAD4 22 1 25 26 34 33 +M00007 +M00007 .050 .050 0.0 0.0 CQUAD4 23 1 26 27 35 34 +M00008 +M00008 .050 .050 0.0 0.0 CQUAD4 24 1 27 28 36 35 +M00009 +M00009 .050 .050 0.0 0.0 CQUAD4 25 1 28 29 37 36 +M00010 +M00010 .050 .050 0.0 0.0 CQUAD4 26 1 29 30 38 37 +M00011 +M00011 .050 .050 0.0 0.0 CQUAD4 27 1 30 31 39 38 +M00012 +M00012 .050 .050 0.0 0.0 CQUAD4 28 1 31 32 40 39 +M00013 +M00013 .050 .050 0.0 0.0 ENDDATA

Listing 3-2. Restart Input File for Modifying the Structured Model of HA146E

Postprocessing of Model Data If you want to change some aspect of an original analysis other than the structural model or the aerodynamic model, the restart can be thought of as a postprocessing operation. The NX Nastran restart capability is particularly powerful in this scenario since the CPU-intensive tasks are only performed in the cold start. As an example, the previous example with the increased plate thickness demonstrates a flutter speed in the 620-640 ft/sec range. Suppose it is desired to study this range more intensively for all the modes using a PK-flutter analysis. Relative to the run of the previous example, this requires changing the FLUTTER Bulk Data entry and the FLFACT entry that previously

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specified reduced frequencies and now must specify velocities. Listing 3-3 shows the restart input data file for this case. Note that the analysis velocities have been input using physically consistent units of in/sec. Furthermore, note that VERSION = 3 is specified on the RESTART command. This is the version that contains results from the successful restart run of the previous example. If there is doubt as to the correct version, the .F04 file of the successful run identifies the version that contains the desired information. RESTART VERSION=3,KEEP ID NXN, HA145E_RST2 TIME 5 $ SOL 145 $ FLUTTER ANALYSIS CEND TITLE = EXAMPLE HA145E: HALF SPAN 15-DEG SWEPT UNTAPERED WING SUBT = PK-METHOD FLUTTER ANALYSIS, DOUBLET-LATTICE AERO LABEL = 0.050 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES SEALL = ALL ECHO = BOTH SPC = 1 $ WING ROOT DEFLECTIONS AND PLATE IN-PLANE ROTATIONS FIXED METHOD = 10 $ MODIFIED GIVENS METHOD OF REAL EIGENVALUE EXTRACTION FMETHOD = 30 $ PK-FLUTTER METHOD SVEC = ALL $ PRINT VIBRATION MODES BEGIN BULK / 54 55 FLFACT 3 7440. 7500. 7560. 7620. 7680. FLUTTER 30 PK 1 2 3 ENDDATA

Listing 3-3. Restart Input File for Modifying the Flutter Method from the Input File Shown in Listing 4-2

Addition of m,k Pairs Modification of the MKAEROi entries represents a special case in that neither the aerodynamic or structural models have changed, but additional aerodynamic analyses are required. Before these calculations are made, a check is done to see if the aerodynamic matrices exist for a particular m and k. If they do, the calculation is skipped. Listing 3-4 shows the input data file required for a restart of Examples HA146D and Frequency Response of BAH Wing to Oscillating Aileron (Examples HA146D and HA146DR) with an additional m,k pair of m = 0.0 and k = 0.1. RESTART VERSION=1,KEEP ID NXN,HA146D_RST TIME 10 $ CPU TIME IN MINUTES SOL 146 $ AEROELASTIC RESPONSE CEND TITLE = BAH WING DYNAMIC FREQUENCY RESPONSE HA146D SUBTI = ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO LABEL = UNIT, HARMONIC AILERON LOADING ECHO = BOTH SPC = 13 $ BOUNDARY CONDITIONS (ANTISYMMETRIC) MPC = 1 $ CONTROL SURFACE RELATIVE MOTION METHOD = 10 $ MODIFIED-GIVENS EIGENVALUE METHOD SDAMP = 2000 $ STRUCTURAL DAMPING (3 PERCENT) DLOAD = 1000 $ FREQUENCY DEPENDENT LOAD FREQ = 40 $ FREQUENCY LIST OUTPUT SET 1 = 11 SPCF = 1 $ SINGLE POINT CONSTRAINT FORCES OUTPUT(XYOUT) $ XY PLOTTING PACKAGE CSCALE 2.1 PLOTTER NASTRAN CURVELINESYMBOL = 0

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Input Files for Aeroelastic Problems

XMIN = 0. XMAX = 5.0 XGRID LINES = YES YGRID LINES = YES UPPER TICS = -1 RIGHT TICS = -1 XTITLE = FREQUENCY (CPS) TCURVE = WING ROOT BENDING MOMENT YTITLE = SPC FORCE OF GRID 11 R3 XYPLOT SPCF / 11(R3) BEGIN BULK MKAERO1 0.0 0.10 ENDDATA

Listing 3-4. Input Data File for Adding a Mach Number and Reduced Frequency Pair to Example HA146D

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Chapter

4 Output Features and Interpretation

•

Overview

•

Static Aeroelasticity

•

Flutter

•

Dynamic Aeroelasticity

•

Design Sensitivity And Optimization

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Chapter 4

4.1

Output Features and Interpretation

Overview

Aeroelastic analysis and design involves four solution sequences (SOLs 144, 145, 146, and 200). Each solution has its own standard output format and each can be supplemented with additional output obtained by appropriate Case Control commands and the use of parameters (PARAMs) in the Bulk Data or Case Control files. The examples of Static Aeroelastic Analysis Sample Problems through Dynamic Aeroelastic Response Analysis provide a comprehensive description of the aeroelastic output so that this chapter is limited to a short description of the output and an indication of where it is described and depicted in the examples.

4.2

Static Aeroelasticity

Stability Derivatives Nondimensional stability derivatives are printed for any static aeroelastic analysis that includes SUPORT degrees of freedom and for as many degrees of freedom as are SUPORTed. FSW Airplane in Level Flight (Example HA144A) provides a description of this output while Listing 6-3, Listing 6-10, et al., provide examples.

HP0 and HP Matrices The HPO and HP matrices are used to determine mean axis rotation angle derivatives that can be used, for example, in time simulation studies that use restrained aeroelastic coefficients in the equations of motion. The number of rows in these matrices correspond to the number of SUPORTed degrees of freedom. The HP0 matrix is for the intercept input (e.g., user input pressures and downwash angles) while the HP matrix contains data for each of the AESTAT and AESURF entries in the same order as they appear in the stability derivative print. FSW Airplane in Level Flight (Example HA144A) provides a description of how to interpret these data for a longitudinal analysis while FSW Airplane in Antisymmetric Maneuvers (Example HA144D) contains a lateral example.

Trim Variables The results of the trim analysis for all the aerodynamic extra points are always printed for each subcase. The values are in the units specified on the AESTAT Bulk Data entry or in radians for AESURF derived extra points. Each of the output listings in Static Aeroelastic Analysis Sample Problems contains at least one example of this output.

Aerodynamic Pressures and Forces This output table contains information on the pressures and forces on the aerodynamic elements at the trimmed flight condition. Both pressure coefficients and pressures are provided and one pressure per element is printed using the numbering convention for the j-set degrees of freedom described at the end of Aeroelastic Modeling. The aerodynamic “forces” are printed out in two columns, corresponding to the forces and moments acting on the panels at the aerodynamic grid points. The numbering convention for these k-set degrees of freedom can also be found at the end of Aeroelastic Modeling. Listing 6-25 contained in FSW Airplaine with Bodies (Example HA144F) is particularly illustrative of this print in that it contains results for bodies as well as lifting surfaces. The body “pressures” are actually singularity magnitudes as defined by Eq. 1-5 and therefore do not have physical significance. Unit Solutions for Loadings of the FSW Airplane

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Output Features and Interpretation

(Examples HA144GA and HA144GB) present options that are available for using this print to obtain results for a particular aerodynamic extra point.

Standard Data Recovery Once a trim analysis has been performed, you can determine structural displacements and then perform standard NX Nastran data recovery to provide output such as element stresses and forces and grid point displacements and SPC forces. See Also •

NX Nastran Output Files in the NX Nastran User’s Guide

Divergence Analysis A divergence analysis produces two tables in the results file. An example of this output is given in Listing 4-1 for the input file discussed in Static Aeroelastic Analysis. The first table is the standard output from a complex eigenanalysis. Only roots that are purely imaginary and positive are physically meaningful. The second table therefore has screened the eigenvalues and prints them out in ascending magnitude of the divergence dynamic pressure, where

Equation 4-1. EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS DIVERGENCE ANALYSIS C O M P L E X E I G E N V A L U E ROOT EXTRACTION EIGENVALUE NO. ORDER (REAL) (IMAG) 1 2 -8.485735E+00 6.882407E-15 2 1 -5.327813E-15 4.548029E+00 3 3 -5.993966E-16 9.788890E+00 4 4 -6.475911E-14 1.791923E+01 5 5 1.594443E-13 2.840956E+01 6 6 1.362681E-13 5.442617E+01 7 7 4.771999E+06 3.794945E+06 8 8 -3.616392E+05 6.943392E+06 EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS DIVERGENCE ANALYSIS EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS DUVERGENCE ANALYSIS D I V E R G E N C E S U M M A R Y MACH NUMBER = .000000 METHOD = COMPLEX LANCZOS ROOT DIVERGENCE EIGENVALUE NO. DYNAMIC PRESSURE REAL IMAGINARY 2 2.068457E+01 -5.327813E-15 4.548029E+00 3 9.582236E+01 -5.993966E-15 9.788890E+00 4 3.210987E+02 -6.475911E-14 1.791923E+01 5 8.071033E+02 1.594443E-13 2.840956E+01 6 2.962208E+03 1.362681E-13 5.442617E+01

PAGE 19 S U M M A R Y FREQUENCY (CYCLES) .0 7.238412E-01 1.557950E+00 2.851933E+00 4.521522E+00 8.662194E+00 6.03984EE+05 1.105075E+06

DAMPING COEFFICIENT .0 2.342911E-15 1.224647E-16 7.227891E-15 -1.122469E-14 -5.007446E-15 -2.514924E+00 1.041679E-01 PAGE 20 PAGE 21

Diagnostic Output Certain NX Nastran modules provide the ability to output additional information using “diagnostic” prints. Two examples of this that are relevant to static aeroelasticity are: •

DIAG 39 print aerodynamic box geometry information for ZONA51 aerodynamic panels inside the AMG module.

•

DIAG 50 prints transformation information inside the ADG module. This was inserted during the development of this module, but is of limited utility to users.

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Output Features and Interpretation

Plots NX Nastran contains an internal plotter that provides minimal support for aeroelasticity. In the current engineering environment of graphical user interfaces, this capability is becoming obsolete. However, since it contains features that are not available in any other product, these features are documented here. For static aeroelasticity, the geometry of the aerodynamic models can be displayed, including the meshing of the aerodynamic panels. This can be plotted in conjunction with the structural model so that it can be invaluable in identifying user input errors. Plotting in the NX Nastran User’s Guide contains information of the use of the internal plotter. The AERO1 element identifies aerodynamic models for plotting purposes. FSW Airplane in Level Flight (Example HA144A) shows the Case Control requests required to produce the plot of the aerodynamic and structural model shown in Figure 4-1.

Figure 4-1. A Plot of the Forward-Swept-Wing (FSW) Airplane Aerodynamic and Structural Models Generated in the Example of Section 7.2 Using the NX Nastran Internal Plotter

4.3

Flutter

Real Eigenanalysis Eigenvalue results from the normal modes analysis are always printed while real eigenvector results are printed if the appropriate Case Control commands are present (SVEC for displacements in the a-set, DISP for global displacements). PARAM,OPPHIPA,1 can be used to output the eigenvectors at the aerodynamic grid points and can therefore assist in assessing the quality of the structural splining.

Flutter Summaries A tabular listing is provided of the flutter results with each of the three flutter methods using a customized format. Listing 7-3 shows output for the PK- and K-methods applied to the same

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Output Features and Interpretation

flutter analysis task. For the PK-method, roots are sorted in increasing frequency. POINT 1 then outputs the lowest frequency root at each of the velocities used in the flutter analysis and subsequent POINT’s refer to the higher frequencies. If there are multiple densities and/or Mach numbers, POINTS corresponding to the first Mach number and density are printed first, followed by the (optional) second Mach number and first density and finishing with the final Mach number and final density. It is relatively easy to scan the damping column of a particular POINT to see if a branch is going unstable. Once a crossing is observed, the flutter velocity and frequency can be interpolated from the data that brackets the crossing. K-method results are a bit more difficult to interpret. For this method, a POINT corresponds to a particular reduced frequency, Mach number, density triplet, with the data within the POINT arranged in increasing values of frequency. In this case, it is necessary to manually trace a flutter branch across POINTs to determine where a root goes unstable. An example of a KE-method flutter summary is shown in Listing 7-16. For this method, a sorting algorithm, described in Eq. 1-121 and Eq. 1-122, is used to arrange the data in flutter branches, much like those already described above for the PK-method. The sorting algorithm is not robust so that sometimes branches become intertwined, but it is usually possible to untangle them using the plotted output of the V-g and V-f curves. PARAM,PRINT,NO suppresses the print of the flutter summary. This can be used when V-g and V-f plots are relied upon to display the results of the flutter analysis.

Flutter Eigenvectors Eigenvectors associated with the flutter eigenvectors are available for the PK and K methods of flutter analysis. An example of this output for the PK-method is given in Listing 7-3. The user indicates that eigenvectors are to be output for a particular velocity by entering a minus sign in front of that velocity on the FLFACT entry. Two sets of eigenvectors are produced, one in modal coordinates (i.e., the eigenvector that is generated as part of the complex modal eigenanalysis) and the second in physical coordinates (i.e., the eigenvector that results when the modal eigenvector is expanded to physical coordinates using the normal modes eigenvectors). The first set appears prior to the flutter summary and is printed as the roots are extracted in the flutter analysis. The second set of eigenvectors is generated and printed following the flutter analysis as a data recovery operation. In both cases, the burden is on the user to connect the eigenvector with the particular flutter point by searching for the eigenvalue of interest. It is also possible to recover other responses, such as element stresses, for the eigenvectors. An example of flutter eigenvectors produced by the K-method is also given in Listing 7-3. The OFREQUENCY Case Control command can be used to restrict the range of frequencies over which eigenvector results are output. Diagnostic Output As in the static aeroelastic description above, DIAG 39 produces aerodynamic box geometry information for ZONA51 aerodynamic modules inside the AMG module. It also provides detailed information on the PK-method of flutter analysis that provides a history of the iterations used to produce the Flutter Summary results. Plots As in static aeroelasticity, the geometry of the aerodynamic models can be displayed, including the meshing of the aerodynamic panels. In addition, the real eigenvectors can be displayed on the aerodynamic model if PARAM,OPPHIPA,1 has been used. Only the VECTOR option can be used for these displacements.

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Output Features and Interpretation

V-g and V-f plots may be requested by the XYOUT Case Control commands by specifying the curve type as VG. The “points” are loop numbers and the “components” are g, structural damping, or f, frequency. An example of these plot requests can be found in Listing 7-5. The first plots produced by this request are displayed in Figure 4-2.

Figure 4-2. A V-g, V-f Plot Generated from the Example Contained in Section 8.3 Using the NX Nastran Internal Plotters

4.4

Dynamic Aeroelasticity

Output for Dynamic Aeroelastic Response Analysis Output, sorted by point number or element number (SORT2), is available either as real and imaginary parts or magnitude and phase angle (0° to 360°) for the list of frequencies specified by the OFREQUENCY Case Control command. For transient problems, real numbers are output at the list of times specified by the OTIME Case Control command.

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Output Features and Interpretation

Transient and Frequency Response The following data may be printed in transient and frequency response outputs analysis: •

Displacements, velocities, and accelerations for a list of physical points (grid points and scalar points introduced for dynamic analysis) or for solution points (points used in formulation of the matrices in the general equations of motion).

•

Nonzero components of the applied-load vector and single-point forces of constraint for a list of grid points and extra points.

•

Aerodynamic pressures and forces on selected aerodynamic elements.

•

Stresses and forces on selected elements (= ALL, which specifies all elements, is available only for SORT2).

Random Response The following printed output is available for random response calculations: •

The power spectral density function and the root mean square value of the response of listed components for points or elements.

•

The expected number of zero crossings with positive slope per unit time, N0.

•

The autocorrelation function of the response of listed components for points or elements.

Plots The following plotter output is available for dynamic aeroelastic response analyses: •

Undeformed plot of the structural model.

•

XY-plot of any component of displacement, velocity, or acceleration of a list of points versus time or frequency.

•

XY-plot of any component of the applied-load vector or single-point force of constraint versus time or frequency.

•

XY-plot of any stress or force component for an element versus time or frequency.

The following additional plotter output is available for random response analyses: •

XY-plot of the power spectral density versus frequency for the response of listed components for points or elements.

•

XY-plot of the autocorrelation versus time delay for the response of listed components for points or elements.

The data specified for the XY-plots may also be punched or printed in tabular form [see Plotting in the NX Nastran User’s Guide]. Also, a summary is printed for each XY-plot that includes the maximum and minimum values of the plotted function.

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4.5

Output Features and Interpretation

Design Sensitivity And Optimization

Solution Sequences in theNX NASTRAN Design Sensitivity and Optimization User’s Guide contains a comprehensive description of the output from SOL 200. For aeroelastic optimization tasks, PARAM,NASPRT can be quite useful. This parameter indicates how frequently results are printed during the design task. Since these results can be extensive, using the default will minimize the number of pages and output that is produced.

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Chapter

5

Aeroelastic Solution Sequences

•

Overview

•

Solution Sequence Functions

•

Solution Sequence Structure

•

Aeroelastic Modules

•

Selected Aeroelastic Data Blocks

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Chapter 5

5.1

Aeroelastic Solution Sequences

Overview

As discussed in Executive Control Section , the standard use of NX Nastran requires a solution sequence, and there are four solution sequences that relate to aeroelasticity: SOL Number 144 145 146 200

SOL Name AESTAT SEFLUTTR SEAERO DESOPT

Description Static Aeroelasticity Aeroelastic Flutter Dynamic Aeroelasticity Design Sensitivity and Optimization

This chapter first briefly describes the functionality of each of these solution sequences. Some users require a more in-depth understanding to the solution sequences; e.g., to extract intermediate results or to alter the solution sequence to provide functionality not provided in the basic sequence. For this reason, the remainder of the chapter provides significant detail on the solution sequences. This includes a description of the key subDMAPS, brief descriptions of each of the modules, and a listing of key data blocks.

5.2

Solution Sequence Functions

Static Aeroelasticity The static aeroelastic solution sequence (SOL 144) provides the following capabilities: •

The user supplies finite element models for the definition of the structure and aerodynamic loading, including information on the flight condition. The loads and accelerations are assumed to be independent of time i.e., quasi-steady.

•

Stability and control derivatives are printed for each unique flight condition (Mach number and dynamic pressure). Derivatives are printed for the rigid vehicle and for the restrained and unrestrained elastic vehicles.

•

A trim analysis is performed that determines unknown trim values and then performs standard data recovery for each TRIM subcase defined in the Case Control section of the input data file. Aerodynamic forces and pressures on the aerodynamic elements may be obtained via the AEROF and APRES Case Control commands, respectively.

•

Three matrices are available for altering the theoretically predicted aerodynamics. Correction factors can be input using WKK, experimental pressures can be input using FA2J and adjustments to the downwash to account for, e.g., the effects of camber and twist, can be input using matrix W2GJ.

•

A static aeroelastic divergence analysis is available by specifying a DIVERG Case Control command in a subcase. The divergence analysis is performed at the Mach numbers specified on the corresponding DIVERG Bulk Data entry.

Any of the aerodynamic methods can be utilized for divergence analysis. Strip Theory, the Mach Box method, and Piston Theory are not available for trim and stability analysis.

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Flutter Analysis The flutter solution sequence (SOL 145) provides a comprehensive flutter analysis with the following capabilities: •

The user supplies finite element models for the definition of the structure and the aerodynamic model. Aerodynamic matrices are computed explicitly at each of the user-supplied Mach number and reduced frequency combinations.

•

A modal analysis is always performed. Changes in the mass and stiffness matrices may be made subsequent to the modal analysis via DMIG Bulk Data entries.

•

Control systems can be modeled using extra point, transfer function and DMIG inputs. The user can supply downwash vectors for extra point motions using the DMI matrices D1JE and D2JE.

•

A flutter analysis is performed based on the parameters specified on the FLUTTER Bulk Data entry that is selected by the FMETHOD Case Control command. The K- and KE-methods compute flutter roots for user-specified values of density, Mach number and reduced frequency. The PK-method extracts these roots for user-specified values of density, Mach number and velocity.

•

Multiple subcases can be specified. This enables the use of, e.g., different flutter solutions or multiple sets of DMIG information.

•

A flutter summary is printed and (optionally) V-g and V-f plots are produced.

•

Data recovery can be performed on the flutter eigenvectors produced for the K- and PK-flutter solutions.

All NX Nastran aerodynamic theories are available. You can include more than one aerodynamic theory in the same aerodynamic model.

Dynamic Aeroelasticity The dynamic aeroelasticity solution sequence (SOL 146) provides analysis capability in the time or frequency domain. The following capabilities are available: •

You supply finite element models for the structure and the aerodynamics. Aerodynamic matrices, including gust loads, are computed at each of the user-specified Mach number and reduced frequency combinations.

•

Frequency or time-dependent loading can be specified. Time varying loads are converted to the frequency domain using ad hoc Fourier transform techniques (see Dynamic Aeroelastic Analysis ). The excitation can be aerodynamic (such as gust loading), or external (such as mechanical loads representing store ejection or landing loads).

•

The software always performs a modal analysis. Changes in the mass and stiffness matrices may be made subsequent to the modal analysis via DMIG Bulk Data entries.

•

Control systems can be modeled using extra point, transfer function, and DMIG inputs. The user can supply downwash vectors for extra point motions using DMI matrices D1JE and D2JE.

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•

Basic computations are always performed in the frequency domain. If input is provided in the time domain, an inverse Fourier transform is used to provide output in the time domain.

•

The modal participation type of data recovery is used. The internal loads or stresses are found in each mode and the response loads are found from the linear combination of the products of the loads in each mode and its amplitude. This method of internal load response calculation is called the “Modal Displacement Method” in Bisplinghoff, Ashley, and Halfman (1955, pp 641-650).

•

Output can be displacements (including velocities and accelerations), stresses, or constraint forces. XY-plots are available. Aerodynamic data (pressures and forces) are also available with frequency response analysis.

•

Random response analysis obtains power spectral density, root mean square response, and mean frequency of zero crossings.

All NX Nastran aerodynamic theories are available for calculating the dynamic aeroelastic response to external loading. The Strip, Mach Box, and Piston Theory aerodynamics are not available for gust loads.

Design Sensitivity and Optimization The design sensitivity and optimization solution sequence (SOL 200) contains multidisciplinary analysis and design capabilities that are beyond the scope of the this guide. The reader is referred to the companion NX Nastran Design Sensitivity and Optimization User’s Guide for a focused discussion of SOL 200. For the purposes of this guide, the aeroelastic capabilities are summarized in this section: •

The full range of static and flutter analysis capabilities of SOLs 144 and 145 are available in SOL 200.

•

Sensitivity of analysis responses with respect to changes in properties of the structural finite element model can be computed. For static aeroelastic analyses, these responses include not only standard displacement and stress responses, but also sensitivities of stability derivatives and trim variables. For flutter, the sensitivity of the damping levels computed in a PK-flutter analysis are available.

•

Multidisciplinary optimization enables the simultaneous consideration of responses from any number of disciplines in order to formulate a structural design that minimizes a user-defined quantity, such as vehicle weight, while satisfying imposed design conditions (e.g., requirements on stress, roll performance, and flutter stability).

Although all of the SOL 144 and 145 analysis capabilities are available in SOL 200, a subset of these capabilities is available for sensitivity and optimization. In static aeroelasticity, the divergences capability is not available. For flutter, only the PK-method of analysis can be used in flutter design.

5.3

Solution Sequence Structure

The intent of this section is to provide an overview of the solution sequences related to aeroelasticity by a discussion of the underlying subDMAPs. It is beyond the scope of this guide to provide a comprehensive treatment of the DMAP listings.

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Structured solution sequences are composed of a main driver subDMAP, which can invoke any number of additional subDMAPs. Figure 5-1, Figure 5-2, and Figure 5-3 show the high-level structure of subDMAPs AESTAT, SEFLUTTR, and SEAERO, which are the drivers for SOLs 144, 145, and 146, respectively. In the figures, items in boxes are subDMAPs while items in ovals are modules. The shaded boxes are the key aeroelastic subDMAPs and are discussed further in this section. It is seen that there is a great deal of commonality among the sequences. The shared, nonshaded subDMAPs include: •

SUPER1 – Performs initial processing extending from input file processing through the assembly of the global matrices.

•

PMLUSET – Set USET parameters.

•

SEGOA – Forms a-set GOA matrix from the t and q submatrices. This is relevant in advanced reduction methods.

•

SUPER3 – Solution vector data recovery.

The following subDMAPs are only used in modal solutions (SOLs 145 and 146): •

OPPH – Processes normal modes for data recovery and plots.

•

GMA – Converts physical matrices into modal coordinates.

•

VDR1 – Processes xy plot and solution set requests.

•

MODACC – Selects vectors for further postprocessing.

•

MODERS – Performs normal modes analysis.

Two modules shown in the figures are: •

DPD – Dynamic pool distributor. Processes dynamics-related input.

•

SSG2 – Matrix reduction module. In these applications, it reduces the spline matrix from the g- to the a-set.

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Aeroelastic Solution Sequences

Figure 5-1. Solution Sequence 144 (Static Aeroelasticity)

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Figure 5-2. Solution Sequence 145 (Aerodynamic Flutter)

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Figure 5-3. Solution Sequence 146 (Dynamic Aeroelasticity) Shaded subDMAPs shown in Figure 5-1 through Figure 5-3 plus aeroelastic sensitivity subDMAPs are identified in Table 5-1 with the remainder of this subsection devoted to discussion of each. The discussions are meaningful only if a listing of the subDMAP is available. A complete solution sequence listing may be obtained making a NX Nastran run with DIAG 14 set in the Executive Control Section of the input data file. This will list the complete solution sequence. If

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Aeroelastic Solution Sequences

only a particular subDMAP, e.g., PFAERO, is of interest, this can be obtained with the following Executive Control Command: COMPILE PFAERO SOUIN=MSCSOU LIST NOREF

Table 5-1. Aeroelastic SubDMAPs Solution Sequence SubDMAPS 144 145 146 200 X X X X PFAERO X AESTATRS X FLUTTER X X MFREQRS X X* DESAERDR

X

SAERSENS

X

Function Performs preface aerodynamics calculations Statics aeroelastic analysis Flutter analysisi Modal frequency analysis Generates output solution vectors for static aeroelastic sensitivity analysis Performs static aeroelastic sensitivity analysis

* The MFREQRS subDMAP in SOL 200 only supports standard modal frequency analysis without including aeroelastic effects.

PFAERO This subDMAP performs all processing of the aerodynamic data that is independent of the structural model, plus it generates the global spline matrix. The steps involved are: 1. Read in any DMI input (DMIIN). 2. Process aerodynamic model geometry (APD). 3. If there are plot requests, the aerodynamic model data is processed and plotted (PLOT). 4. Print USET data, if requested (TABPRT). 5. Form spline matrix (GI). 6. Determine which aerodynamic methods are present. 7. If only unsteady results are required, go to step 20. 8. Form static aeroelastic matrices that are only a function of geometry (ADG). 9. Loop on the number of static aeroelastic subcases. 10. Loop on the number of Mach numbers per subcase. For trim analysis, this number is one. For divergence analysis, this is the number of Mach numbers appearing on the DIVERG Bulk Data entry. 11. Determine Mach number of the current pass (AELOOP). 12. If the aerodynamic data are already present for this Mach number, go to step 18. 13. Generate aerodynamic matrices (AMG). 14. Apply weighting factors, if any. [WSKJ] = [WKK][SKL].

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15. For divergence analysis, it is possible to mix aerodynamic methods. For brevity, the description of the portion of the subDMAP that enables this capability is not provided. 16. Form: [QKKS] = [WSKJ][AJJ]−1[DJK]. 17. Form: [QKX] = [WSKJ][AJJ]−1[DJK]. 18. Next Mach number. 19. Next subcase. 20. If only steady results are required, go to step 36. 21. Determine if aerodynamic matrices are to be generated (PARAML). 22. Loop on the number of Mach number, reduced frequency pairs. 23. Determine Mach number and reduced frequency. 24. If the required aerodynamic data are already present for this condition, go to step 35. 25. Compute aerodynamic matrices (AMG). 26. Apply weight factors, if any. [SKJ1] = [WTFACT][SKJF]. .

27. Form:

28. It is possible to mix aerodynamic methods. For brevity, the lengthy DMAP code that enables this capability is not described. 29. If Strip Theory, Mach Box method, or Piston Theory, go to step 34. 30. Form: [QKJ] = [AJJ]−1 [SKJ]T. 31. Form: [QKK] = [QKJ][DJKB]. 32. If there are user input downwash matrices due to extra points, form . 33. Go to step 35. 34. Form: [QKK] = [SKJ1][AJJT][DKJB]. 35. Process the next Mach number and reduced frequency. 36. Return.

AESTATRS This subDMAP performs the basic static aeroelastic operations as discussed in Static Aeroelasticity. The steps involved are: 1. Perform various matrix operations using the rigid body matrix, D, and partitions of the mass matrix:

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2. Form invariant matrix [RFSOP] = [TR]T [TRX]. 3. Begin loop on the number of Case Control records. Return when there are no more records. 4. If this is not a TRIM subcase, go to step 27. 5. Obtain Mach number and dynamic pressure for the current subcase (AELOOP). 6. If data already exist for this Mach number and dynamic pressure, go to step 12. 7. Perform matrix operations to obtain aeroelastic stiffness matrices:

8. If there are SUPORT degree(s) of freedom, partition the [KSAA1], [KALX], and [KAAA] matrices into the l-set and r-set components. 9. Form: [ALX] = [KSALL]−1 [KALX]. 10. If there are no SUPORT degree(s) of freedom, go to step 12. 11. There are a large number of multiply and adds (MPYAD and SMPYAD), decompositions (DECOMP) and forward/backward substitutions (FBS) to obtain the matrices required for stability derivative prints and trim analysis. The matrix algebra is provided in Static Aeroelasticity with the notation similar to that used in the subDMAP. The following matrices that result from these calculations are of interest: KRZX Z1ZX RSTAB KSAZX ZZX

= = = = =

Restrained elastic dimensional derivatives Unrestrained elastic dimensional derivatives Rigid, splined dimensional derivatives Rigid, splined dimensional derivatives Solution matrix for trim calculations Perturbations in support point deformations relative to mean axes due to HP = aerodynamic extra points

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12. The MATMOD utility is used to bring in a single column from matrices W2GJ, FA2J, and PA that are used in developing intercept coefficients: = = =

WGJ FAJE PSA

User input downwash vector User input pressure coefficient vector External loads vector

13. Further matrix algebra, also discussed in Static Aeroelasticity , is used to obtain: PZ IPZ IPZF2 RINT INTZ

= = = = =

HP0

=

Loads for the trim calculations Restrained elastic dimensional intercepts Unrestrained elastic dimensional intercepts Rigid, unsplined dimensional intercepts Rigid, splined dimensional intercepts Perturbation in the support point deformations relative to mean axes due to external loads

14. Stability derivatives are printed (SDP). 15. Mean axis deformations are printed. 16. Trim analysis is performed, and the UX vector is computed (ASG). 17. Vehicle accelerations are recovered. 18. l-set displacements are recovered. 19. Aerodynamic grid point displacements due to the structural deformations are recovered:

20. Total downwash velocities are computed:

21. Pressures on the aerodynamic elements are computed:

22. Forces on the aerodynamic elements are computed:

23. Forces are transferred to the omitted degrees of freedom. (This is used in sensitivity analysis.) 24. Forces and pressures are written to the results file (ASDR). 25. Results for the single subcase are appended onto previous subcase results.

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26. Go to step 28. 27. Increment the subcase counter. 28. Return to step 3 for the next subcase. 29. Equivalence data blocks so that they will be retained on the database. 30. Return.

DIVERGRS This subDMAP performs static aeroelastic divergence analysis. The steps involved are: 1. A matrix transform is performed on a spline matrix: [GKL] = [GTKL]T 2. A loop is begun on the number of subcases. 3. Determine if this is a divergence subcase (PARAML). 4. If there are no more subcases, RETURN. If this is not a divergence subcase, go to step 16. 5. Begin a loop on the number of Mach numbers found on the DIVERG Bulk Data entry for this subcase. 6. Extract the Mach number (AELOOP). 7. Check for the presence of the [QLL]matrix for this Mach number. 8. If the [QLL] matrix is not present, form [QLL] = [GKL]T[QKK][GKL]. 9. Determine the eigenvalues and eigenvectors of [KLL + λQLL] (CEAD). 10. Form [QLL]T and repeat the complex eigenanalysis. 11. Print the eigenvalues of the complex eigenanalysis. 12. Print any user-requested right-hand eigenvectors. 13. Print the divergence results. 14. Append results from this subcase to previously computed results. 15. Go to step 17. 16. Increment the subcase counter. 17. Go to step 2. 18. Copy results to the database. 19. Return.

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FLUTTER The FLUTTER subDMAP performs flutter analysis. The subDMAP is called inside a subcase loop from the SEFLUTTR main driver for SOL 145. Steps 17 through 41 listed below describe the FLUTTER subDMAP while the remaining steps describe the subcase do-loop within SEFLUTTR that begins with the statement “DO WHILE (NSKIP >=0)”: 1. Increment SUBCNTR by 1. 2. NSKIP is set to -1 if this is the last subcase (CASE). 3. Obtain parameters for this subcase (PVT). 4. If this is a data recovery only run (EXTRCV>0), go to step 45. 5. Extract parameters that indicate whether the subcase has transfer function or DMIG inputs (PARAML). 6. If this is the first subcase, go to step 10. 7. Determine if the generalized matrices are available for the current parameters (DBSTATUS). 8. If the any of the generalized structural matrices are not available, delete all of them (DELETE). 9. Delete solution vectors. 10. Generate the generalized modal matrices (excepting aerodynamics) (subDMAP GMA). 11. If this is the not the first subcase, go to step 16. 12. Call the PFAERO subDMAP. 13. Reduce the global spline matrix to the a-set (SSG2). 14. Print the spline matrix, if requested (MATGPR). 15. If requested, call the OPPH subDMAP to print/plot the normal modes eigenvectors, including displacements at the aerodynamic grid points. 16. Call the FLUTTER subDMAP. 17. If this not the first subcase, go to step 32. 18. If there are extra points, partition out portions of the PHDH matrix that correspond to extra points. 19. Form: [GPKI] = [GKA][PHAI]. 20. Determine which aerodynamic methods have been used (PARAML). 21. Determine if generalized matrices are to be generated (PARAML). 22. Loop on the number of Mach number, reduced frequency pairs. 23. Set Mach number and reduced frequency qualifiers.

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24. Form: [QKI] = [QKK][GPKI] [QII] = [GPKI][QKI]. 25. If there are no extra points, equivalence [QKI] and [QII] to [QKH] and [QHH], respectively, and go to step 29. 26. Form: [QIE] = [GPKI][QKE]. 27. Form [QHH] by merging [QII] with [QIE] 28. Form [QKH] by merging [QKI] with [QKE]. 29. Append [QHH] and [QKH] onto [QHHA] and [QKHA], respectively (special append feature of module SDR1). 30. End of Mach number, reduced frequency loop. 31. Generate matrix lists [QHHL] and [QKHL] (MATMOD, option 22) 32. Loop on the flutter analysis triplets. 33. For the PK- or KE-methods, perform the required flutter analyses. For the K-method, set up the matrices for the complex eigensolver (FA1). 34. For the PK- and KE-methods, equivalence FA1 outputs [KXHH] and [BXHH] to eigenvector and eigenvalue data blocks, respectively, and go to step 36. 35. For the K-method, perform the complex eigenanalysis (CEAD). 36. Transfer the flutter results to output data blocks (FA2). 37. Go to step 32. 38. Copy data to the database. 39. If requested, form V-g and V-f plots (XYTRAN AND XYPLOT). 40. If requested, print out aerodynamic forces for the flutter eigenvectors (ADR). 41. Return to SEFLUTTR. 42. If there are no flutter eigenvectors, go to step 46. 43. Go to subDMAP VDR1 to process any solution set requests. 44. Go to subDMAP MODACC to extract out eigenvectors requested for further processing. 45. Go to subDMAP SUPER3 to process user requests for eigenvectors in physical degrees of edom. 46. If NSKIP is positive, go to step 2. 47. END.

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MFREQRS This subDMAP performs modal frequency response analysis. For this reason, it is also called by the modal frequency response solution (SOL 111) and is therefore not a purely aeroelastic subDMAP. However, because it is the key subDMAP for SOL 146, it is described here. The subDMAP consists of the following steps: 1. Generate external loads in the frequency domain (FRLG). 2. If there are extra points, partition out portions of the PHIDH matrix that correspond to extra points. 3. Form: [GPKI] = [GKA][PHIA]. 4. Determine if generalized matrices are to be generated (PARAML). 5. Loop on the number of Mach number, reduced frequency pairs. 6. Set Mach number and reduced frequency qualifiers. 7. Form: [QKI] = [QKK][GPKI] [QKK] = [GPKI]T[QKI] 8. if there are no extra points, equivalence [QKI] and [QII] to [QKH] and [QHH], respectively, and go to step 12. 9. Form: [QIE] = [GPKI]T[QKE] 10. Form [QKH] by merging [QII] with [QIE]. 11. Form [QKH] by merging [QKI] with [QKE] 12. Append [QHH] and [QKH] onto [QHHA and [QKHA, respectively, using the special append feature of SDR1. 13. End of Mach number and reduced frequency loop. 14. Generate matrix lists [QHHL and [QKHI] (MATMOD, option 22) 15. Form the loading in generalized coordinates (GUST). 16. Calculate the frequency response in generalized coordinates (FRRD2). 17. If required, transfer the frequency response data into the time domain (IFT). 18. Satisfy user requests for aerodynamic forces on the aerodynamic elements (ADR). 19. Return.

DESAERDR This subDMAP performs specialized matrix operations required in the computations of static aeroelastic sensitivities. The output of the subDMAP is three a-set size matrices that contain (1) all the displacement vectors required in performing aeroelastic sensitivity analysis (AUADS), (2) all the acceleration vectors required in performing aeroelastic sensitivity analysis (AAADS),

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and (3) the subset of the displacement vectors that is required in the sensitivity analysis for unrestrained stability derivatives (UUADS). The steps in the subDMAP are: 1. Matrix [RFSOP] = [TR]T[TRX] is formed and counters for the number of columns in AUADS (NSOL) and UUADS (NMSOL) are initialized to zero. 2. A loop on the number of records in the Case Control data block and that terminates at step 20 is initiated. 3. A determination is made as to whether the sensitivity is required for static responses (STFLG > 0), trim responses (TFLG > 0) and/or stability derivatives (SDFLG > 0). For SDFLG > 0, [UXU] contains pseudotrim vectors for determining unrestrained stability derivative sensitivities, and UXR contains pseudotrim vectors for determining restrained stability derivative sensitivities (DSARLP). 4. If none of the above flags is set, go to step 2. 5. The Mach number and dynamic pressure qualifiers are set for the subcase. 6. Form: [ALX] = [KSALL]−1[KALX]. 7. If STFLG = 0 and TFLG = 0, go to step 11 8. Extract the a-set accelerations and the l-set displacements vector for the current subcase from the respective matrices created in AESTATRS (MATMOD). 9. Expand the displacement vector to the a-set (UMERGE). 10. Append the vector onto the scratch output data blocks and increment NSOL by one. 11. If [UXU] is not present, go to step 16. 12. Increment NMSOL and NSOL by the number of columns in [UXU]. 13. Repeat a number of analysis calculations to create data blocks required for the sensitivity analysis. See the Unrestrained Stability Derivatives for a description of these operations. 14. Recover a-set displacements [UAP] and accelerations [UADDP] for the [UXU] vector based on

[UAP] is formed by a merge of [URP] and [ULP]. 15. [UAP] is appended onto both [AUADSX] and [UUADSX]. [URDDP] is appended onto [AAADSX]. 16. If [UXR] is not present, go to step 20. 17. Increment NSOL by the number of columns in [UXR].

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18. Recover a-set displacements [UAR] and accelerations [AAR] for the [UXR] matrix based on:

[AAR] is formed by a merge of [ALR] and [AAR]

[UAR] is formed by a merge of [ULR] and a purged matrix representing zero displacement in the r-set. 19. [UAR] is appended onto [AUADSX] while [AAR] is appended onto [AAADSX]. 20. Go to step 2. 21. Scratch output data blocks are equivalenced to data blocks stored on the database. 22. Return.

SAERSENS This subDMAP creates sensitivities of static aeroelastic displacements [AULDS], trim variables [DELX], and stability derivatives [DELS]. The steps involved are: 1. The rigid body matrix [DALR] and [RFSOP] matrix are formed. 2. [NCOL], the number of displacement and acceleration vectors, [NMCOL], the number of pseudodisplacement vectors, and [NDV], the number of design variables, are determined. 3. A loop that terminates at step 35 is initiated on the number of records in the Case Control data block. 4. A determination is made as to whether sensitivity analysis is required for static responses (STFLG > 0), trim responses (TFLG > 0)), and/or stability derivatives (SDFLG > 0). For SDFLG > 0, [UXU] contains pseudotrim vectors for determining unrestrained stability derivative sensitivities, and [UXR] contains pseudotrim vectors for determining restrained stability derivative sensitivities (DSARLP). 5. if STFLG and TFLG are both zero, construct a null matrix all with l-size rows and NDV columns and append it to LAULAX. 6. If all of the DSARLP flags are zero, go to the next subcase (step 3 ). 7. Set Mach number and dynamic pressure qualifiers. 8. Form: [KAZR] = [DM]T[KALR] + [KARR]. 9. If [STFLG] and [TFLG] are zero, go to step 16. 10. Create a partitioning vector to cut out the required pseudoloads from the [PA] matrix (MATGEN, option 4).

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11. Partition out the vectors into matrix [DPSAA] and equivalence the remaining columns to [PA]. 12. Partition the a-set loads to the l-set and solve for the pseudodisplacements:

13. Create trim pseudoloads:

14. Solve for [DUX], the perturbed trim variables (ASG). 15. Recover [DUL], sensitivity displacements in the l-set:

16. The sensitivity vectors are appended onto a scratch output data block. 17. If [UXU] is absent, go to step 23. 18. Recreate some AESTATRS data blocks that were not stored (see Static Aeroelasticity ). 19. Create a partitioning vector to cut out the required pseudoloads from the [PA] matrix (MATGEN, option 4). 20. Partition out the vectors into matrix [DPSAAU] and equivalence the remaining columns to [PA]. 21. Create a partitioning vector to cut out the required pseudoloads due to mean axis deformations from the PMA matrix (MATGEN, option 4). 22. Partition out the vectors into matrix [DPSAUM] and equivalence the remaining columns to [PMA]. 23. A series of matrix algebra statements is now executed to form matrix [DSTABU], a matrix of dimensional unrestrained stability derivative sensitivities. Static Aeroelastic Sensitivity contains a description of these operations in the notation used in this subDMAP. 24. If [UXR] is absent, go to step 28. 25. Create a partitioning vector to cut out the required pseudoloads from the [PA] matrix (MATGEN, option 4). 26. Partition out the vectors into matrix [DPSAR] and equivalence the remaining columns to [PA]. 27. Remove the r-set degrees of freedom from [DPSAR] with the result placed into [PLSTBL].

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28. Form the dimensional restrained stability derivative matrix:

29. The information contained in matrices [DUX], [DSTABR], and [DSTABU] is extracted and converted into matrices [DELX1] and [DELS1]. [DELX1]contains sensitivities for the user-requested trim variables while [DELS1] contains sensitivities for all the user-requested stability derivatives. (DSARSN) 30. If this is the first subcase to generate [DELS1], equivalence [DELS1] to [DELS2] and go to step 31. 31. Merge matrices [DELS1] ] and [DELS2] ] into [DELSX] and equivalence [DELSX] to [DELS2]. 32. If this is the first subcase to generate [DELX1], equivalence [DELX1] to [DELX2] and go to step 33. 33. Merge matrices [DELX1] and [DELX2] into [DELXX] and equivalence [DELXX] to [DELX2]. 34. Go to step 3. 35. Scratch data blocks [DELX2] and [DELS2] are equivalenced to NDDL data blocks [DELX] and [DELS]. 36. Scratch data block [LAULDX] contains sensitivity vectors for multiple subcase and multiple design variables. Further processing of these vectors require that all the vectors for a given design variable be contiguous. A do loop on the number of design variables rearranges the vectors that have all vectors for a given subcase contiguous into the required order and places the results in scratch data block [AULDSX]. 37. [AULDSX] is equivalenced to NDDL data block [AULDS]. 38. Return.

5.4

Aeroelastic Modules

The following modules are unique to aeroelasticity. A detailed description for most of them can be found in the NX Nastran Programmer’s Manual. ADG

The Aerodynamic Downwash Generator calculates the downwash matrix, which specifies the downwash for each of the aerodynamic extra points. It also forms matrices required in the generation of stability derivative information and in the specification of the aerodynamic trim equations.

ADR

Aerodynamic Data Recovery builds a matrix of aerodynamic forces per frequency for each aerodynamic point. The data are output for a user-selected set.

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AELOOP

Aeroelastic Loop extracts a single record of Case Control and sets up values that drive the subsequent DMAP instructions for the generation of aerodynamic matrices and/or the performance of static aeroelastic analyses.

AMG

The Aerodynamic Matrix Generator generates aerodynamic influence matrices (AJJT) and the transformation matrices needed to convert these to the interpolated structural system (SKJ, D1JK, D2JK).

APD

The Aerodyanmic Pool Distributor generates aerodynamic “boxes” for all aeroelastic solutions. Tables are assembled to account for the “box” coordinates.

ASDR

The Aeroelastic Static Data Recovery module prints the aerodynamic extra point displacements and the aerodynamic pressures and forces as requested in the Case Control Section.

ASG

The Aerodynamic Solution Generator solves for the aerodynamic extra point displacements.

DIVERG

The Divergence module determines which of the complex eigenvalues extracted in CEAD are physically meaningful and performs a formatted print of the divergence information. It performs a partition of the eigenvectors, saving the eigenvectors that correspond to the divergence roots.

DSARLP

This module calculates pseudodisplacements that are to be used in the calculation of the sensitivities of stability derivatives. It also determines parameters that are required for all static aeroelastic sensitivity analyses.

DSARSN

Calculates and stores delta response values for trim variables and stability derivatives.

DSFLTE

This module calculates right and left eigenvectors for eigenvalues that have been extracted in a flutter analysis and have been flagged for sensitivity analysis.

DSFLTF

This module calculates the sensitivity of active flutter responses.

FA1

The FA1 module prepares the modal matrices MXHH, BXHH, and KXHH for the K-method of eigenvalue analysis or does the complete eigenvalue analysis for the KE- or PK-method.

FA2

The FA2 module collects aeroelastic flutter data for reduction and presentation for each triplet of the configuration parameters.

GI

The Geometric Interpolation module generates the transformation matrix from structural to aerodynamic displacements.

IFT

The Inverse Fourier Transform module obtains solutions as a function of time for aeroelastic problems for which the aerodynamic forces are only known as functions of frequency.

SDP

The Stability Derivative Printer module calculates and prints the nondimensional stability and control derivatives.

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Aeroelastic Solution Sequences

Selected Aeroelastic Data Blocks

The data blocks listed in this section are of particular interest in aeroelasticity. These data blocks have been chosen from a much larger set because they contain the key data that are most likely to be of interest to the user. A complete description of most of the data blocks listed here can be found in NX Nastran Programmer’s Manual. The module that creates the particular data block is noted at the end of each brief description.

Aerodynamic Model and Spline Information ACPT

Aerodynamic Connection and Property Table. Contains CAEROi, PAEROi, and AEFACT inputs after they have been preprocessed (APD).

AERO

Aerodynamics matrix generation data. Contains AERO and MKAEROi inputs after they have been preprocessed (APD).

EDT

Element Deformation Table. This table contains all Bulk Data entries related to aeroelasticity (IFP).

GTKG

Aerodynamic transformation matrix. {uk } = [GTKG] {uk }, {Fg } = [GTKG]T {F}. The matrix has as many rows as there are degrees of freedom in the g-set and as many columns as there are degrees of freedom is the k-set (GI).

SPLINE

Contains SPLINEi inputs after they have been preprocessed (APD).

Static Aeroelastic Geometry DJX

Downwash matrix. Downwash at the aerodynamic grid points due to motion of an aerodynamic extra point (ADG).

SRKT

Aerodynamic summation matrix. Sums forces acting at the aerodynamic degrees of freedom to the aerodynamic reference point (ADG).

TR

Transformation matrix. Transforms forces from the SUPORT degrees of freedom to the aerodynamic reference point (ADG).

TRX

Acceleration selection matrix. A Boolean matrix to select accelerations from the list of aerodynamic extra points (ADG).

XLIST

Extra point list table. Contains AESTATRS and AESTAT inputs after they have been preprocessed (ADG).

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Aerodynamics Matrices AJJT

Aerodynamic influence coefficient matrix. For Doublet-Lattice and ZONA51 aerodynamics, the transpose of this matrix computes the downwash on the aerodynamic elements for a specified pressure vector. For Strip, Mach Box, and Piston Theories the matrix computes the pressure on the aerodynamic elements for a specified downwash vector. This matrix is for unsteady flows and is therefore complex (AMG).

D1JK

Real part of the substantial differentiation matrix. This matrix, when combined with D2JK, computes the downwash at the aerodynamic control point due to deflections of the aerodynamic degrees of freedom. (AMG).

D2JK

Imaginary part of the substantial differentiation matrix (see D1JK above). This matrix is not required for the static aeroelastic analysis (AMG).

QKJ

Gust aerodynamic influence coefficient matrix. Computes forces on the degrees of freedom in the k-set due to downwashes at the aerodynamic control points. This matrix is for unsteady flows and is therefore complex (PFAERO subDMAP).

QKK

Aerodynamic influence coefficient matrix. Computes forces on the degrees of freedom in the k-set due to displacements at the k-set degrees of freedom. This matrix is for unsteady flows and is therefore complex (PFAERO subDMAP).

QKKS

Steady aerodynamic influence coefficient matrix. Computes forces on the degrees of freedom in the k-set due to displacements at the k-set degrees of freedom. This matrix is for steady flows and is therefore real (PFAERO subDMAP).

QKX

Steady aerodynamic influence coefficient matrix for aerodynamic extra points. Computes forces on the degrees of freedom in the k-set due to displacements of the aerodynamic extra points. This matrix is for steady flows and is therefore real (PFAERO subDMAP).

RAJJT

Same as AJJT for steady aerodynamics. This matrix is real (AMG).

SKJ

Integration matrix. Computes forces on the k-set degrees of freedom due to pressures at the aerodynamic control points. This matrix is for steady flows and is therefore real (AMG).

SKJF

Same as SKJ for unsteady aerodynamics. This matrix is for unsteady flows and is therefore complex (AMG).

Static Aeroelastic Solution FFAJ

Vector of aerodynamic pressures on the aerodynamic control points for the trimmed air vehicle (AESTATRS subDMAP).

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LSALL/USALL

These two matrices are upper and lower factors from the decomposition of after it has been reduced to the l-set. They are used repeatedly in the computation of results for static aeroelasticity (AESTATRS subDMAP).

PAK

Vector of aerodynamic forces on the aerodynamic grid points for the trimmed air vehicle (AESTATRS subDMAP).

STBDER

Stability derivative table (SDP).

UX

Trim variable vector. Contains the values of the aerodynamic trim variables for the trim condition specified on the TRIM Bulk Data entry (ASG).

Generalized Aerodynamic Matrices The following matrices are required in the modal aeroelastic solutions. The QHJ matrices are only required for gust analysis. All the matrices are complex. QHJ

Generalized aerodynamics matrix with as many rows as there are aerodynamic control points and as many columns as there are retained modes (FLUTTER and MFREQRS subDMAPs).

QHJL

A “list” of QHJ matrices for all the user-requested Mach numbers and reduced frequencies. The total number of columns in the matrix is therefore the number of retained modes times the number of (m,k) pairs (FLUTTER and MFREQRS subDMAPs).

QHH

Generalized aerodynamics matrix with as many rows and columns as there are retained modes (FLUTTER and MFREQRS subDMAPs).

QHHL

A “list” of QHH matrices for all the user-requested Mach numbers and reduced frequencies. The total number of columns in the matrix is therefore the number of retained modes times the number of (m,k) pairs (FLUTTER and MFREQRS subDMAPs).

Flutter Solution FLAMA

Table of flutter eigenvalues (FA2).

FPHH

Matrix of flutter eigenvectors (FA2).

OVG

Output flutter curve (V-G and V-F) requests (FA2).

Dynamic Aeroelasticity Solution PHF

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External loads in generalized coordinates. For dynamic aeroelastic analysis, this matrix contains any loads other than the ones due to aerodynamic gusts. There are as many rows as there are retained modes and as many columns as there are frequencies in the analysis (FRLG).

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PHF1

Dynamic aeroelastic loads in generalized coordinates. This matrix is the summation of PHF and any aerodynamic gust loads. There are as many rows as there are retained modes and as many columns as there are frequencies in the analysis (GUST).

PKF

Matrix of aerodynamic forces at user-requested frequencies. The forces are provided at the user-requested (AEROF) aerodynamic degrees of freedom for the requested frequencies (ADR).

UHF

Matrix of generalized displacement results from the frequency response analysis. There are as many rows as there are retained modes and as many columns as there are frequencies in the analysis (FRRD2).

UHVT

Matrix of generalized displacement results in the time domain. This is an Inverse Fourier Transform of the UHF matrix and is computed when the user has supplied input, and requires output, in the time domain. There are as many rows as there are retained modes and as many columns as there are time steps in the user request time for displacement, velocity, and acceleration vectors (IFT).

Aeroelastic Design Sensitivity Input Data in the NX NASTRAN Design Sensitivity and Optimization User’s Guide lists the key data blocks for design sensitivity and optimization. The following are additional selected data blocks that are unique to the aeroelasticity design sensitivity calculations: DELFL

A table containing the flutter sensitivity data that is transferred to the DSCM matrix (DSFLTN).

DELS

A table containing stability derivative sensitivity data that is transferred to the DSCM matrix (DSARSN).

DELX

A table containing trim variable sensitivity data that is transferred to the DSCM matrix (DSARSN).

VTQU

A table containing scalars required in performing the flutter sensitivity calculation (DSFLTE).

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6 Static Aeroelastic Analysis Sample Problems

•

Overview

•

FSW Airplane in Level Flight (Example HA144A)

•

Jet Transport Wing in Roll (Example HA144B)

•

A 15-Degree Sweptback Wing in a Wind Tunnel (Example HA144C)

•

FSW Airplane in Antisymmetric Maneuvers (Example HA144D)

•

FSW Airplane in Unsymmetric Quasi-Steady Maneuvers (Example HA144E)

•

FSW Airplane with Bodies (Example HA144F)

•

Unit Solutions for Loadings of the FSW Airplane (Examples HA144GA and HA144GB)

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Static Aeroelastic Analysis Sample Problems

Overview

Static aeroelastic problems consider the application of steady-state aerodynamic forces to a flexible vehicle, which deflects under the applied loads resulting in perturbed aerodynamic forces. The solution of these problems assumes that the system comes to a state of static (or quasi-static) equilibrium. The aerodynamic load redistribution and consequent internal structural load and stress redistributions can be used for design purposes by structural analysts. The possibility of a static aeroelastic instability, i.e., divergence, is also of concern to structural analysts. Static aeroelastic effects are of concern to other analysts as well. For example, aerodynamicists are concerned with the effects on induced drag, and control systems analysts are concerned with the effects on control effectiveness and static stability. The needs of these analysts from several related disciplines have been considered in the static aeroelastic capability of NX Nastran. Seven quasi-static examples are included in this chapter. These examples produce the symmetric and antisymmetric static stability derivatives as well as loads and stresses due to a variety of potential design conditions: •

Examples HA144A and HA144D are symmetric and antisymmetric models of an idealized forward swept wing (hereafter referred to as FSW) configuration.

•

Example HA144E is a full-span model of the FSW configuration.

•

Example HA144F adds a fuselage and two underwing stores to the HA144E example.

•

Examples HA144GA and HA144GB use the FSW configuration to provide unit solutions for the loadings for the initial incidence and each of the trim variables.

•

Example HA144B produces static aeroelastic results for the BAH wing [the jet transport wing analyzed throughout Bisplinghoff, Asheley, and Halfman(1955)].

•

Example HA144C considers a 15-deg swept untapered wing when mounted on a wind-tunnel wall at a prescribed angle of attack.

6.2

FSW Airplane in Level Flight (Example HA144A)

The first example is the FSW airplane considered by Rodden and Love (1985) in trimmed level flight. The model is extremely idealized as shown in Figure 6-1. The wing has an aspect ratio of 4.0, no taper, twist, or camber, but an incidence of 0.1 deg relative to the fuselage, and a forward sweep angle of 30 deg. The canard has an aspect ratio of 1.0, no taper, twist, camber, incidence, or sweep, and is hinged about its quarter-chord. The chords of both the wing and canard are 10.0 ft, the reference chord is chosen as = 10 ft, and the reference area is S = 200 sq ft for the half-span model. Both subsonic (m = 0.9) and supersonic (m = 1.3) speeds are considered.

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Figure 6-1. Idealization of FSW Configuration The half-span model of the wing is divided into 32 equal aerodynamic boxes, as shown on the left wing in Figure 6-1, for both the Doublet-Lattice and ZONA51 methods of aerodynamic analysis, and the canard is divided into eight equal boxes, as also shown in Figure 6-1. Aerodynamic forces on the fuselage are neglected. (Note that the right-hand side is modeled; the aerodynamic boxes are shown on the left side for convenience.) The right wing and fuselage in Figure 6-1 show the structural idealization. Four weights are located at the one-quarter and three-quarter span and chord positions of the wing, and are assumed to be connected to the 50% chord elastic axis by rigid streamwise bars. The weights are 600 lb forward and 400 lb aft, giving a wing centroid at 45% of the wing chord. The wing is assumed to be uniform with equal bending (ELy ) and torsion (GJ) stiffnesses of 25 × 107lb-ft2 and is connected to the fuselage at its root. The right-side fuselage is assumed to have the same bending stiffness as the wing and is shown with four equal and equidistant weights (1500 lb each per side). The fuselage length is 30.0 ft. The total weight per side is 8000 lb, the center of gravity is 12.82 ft forward of the intersection of the fuselage and wing elastic axis, and the centroidal moment of inertia in pitch per side is Iy = 892,900 lb-ft2. For the subsonic case, the airplane is assumed to be flying at a Mach number m = 0.9 at sea level ( = 1200 psf). The low speed characteristics (but at m = 0.9) are obtained by assuming a low value of dynamic pressure, = 40 psf, to illustrate the behavior of the quasi-rigid vehicle. In the supersonic case, the airplane is assumed to be flying at m = 1.3 at 20,000 ft ( = 1151 psf).

Structural Model The input of structural data is considered first. The fuselage model is illustrated in Figure 6-1. The fuselage length from GRID 97 to GRID 100 is 30.0 ft. BAR elements are used between grid points, and a CONM2 weight of 1500 lb is at each fuselage grid point except GRID 90. The wing input is also illustrated in Figure 6-1. Grid points 111, 112, 121, and 122 are connected to the elastic axis by rigid bars. The wing stiffnesses were assumed to be equal in bending and torsion, ELy = GJ = 25.0 + 07; thus, assuming E = 1.44 + 0.9 psf and G = 5.40 + 08 psf, leads to Iy = 0.173611 ft4 and J = 0.462963 ft4, respectively. Values of cross-sectional area, A = 1.5 ft2,

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Static Aeroelastic Analysis Sample Problems

and chordwise inertia, Iz = 20 ft4, are chosen arbitrarily. A nominal symmetrical rectangular cross section with a 6.0 ft chord and 1.0 ft depth is also assumed for the wing structural box for stress recovery purposes at the four corners. The wing forward CONM2 weights are 600 lb, and the aft weights are 400 lb. The half-fuselage material properties are assumed to be the same as in the wing with the same vertical cross-sectional moment of inertia, Iy = 0.173611 ft4. The remaining fuselage cross-sectional area properties are selected arbitrarily for stiffness and stress recovery, specifically, A = 2.0 ft2, Iz = 0.15 ft4, J = 0.5 ft4, and the points selected for stress recovery are at y,z = ±1.0, ±1.0. There are two rigid body motions in this model: vertical translation and rotation in pitch. A SUPORT Bulk Data entry defines a reference point for these rigid body modes on GRID 90, DOFs 3 and 5. Component 4 (roll) of wing grid points 110 and 120 is omitted from the calculation in order to illustrate this means of reducing the problem size and thus has no effect on the results. GRID 90 is constrained longitudinally, and all of the fuselage grid points are constrained for symmetry using SPC1 entries. PARAM entries select GRID 90 as the inertial property reference point and convert the input weights to masses in slugs. [Note that PARAM,WTMASS,1/G provides the conversion of weight to mass; PARAM,AUNITS,1/G allows for the input of the accelerations using load factors (Gs).] CORD2R 100 provides the NACA reference axes for the stability derivatives. The trim angle of attack is the angle of attack of the structural axis at the SUPORT point.

Aerodynamic Model The Doublet-Lattice and ZONA51 methods for surfaces are specified on the CAERO1 entries. CAERO1 1000 specifies the canard with a 2 × 4 division into boxes. CAERO1 1100 specifies the wing with an 8 × 4 division into boxes. The PAERO1 entry is required even though the fuselage modeling is being neglected. Additional aerodynamic data are included in DMI entries to account for the differences between test and theory [the correction factors Wkk of Eq. 1-21], experimental pressure data at some reference condition, e.g., zero angle of attack, [the additive coefficients fe also of Eq. 1-21], and any initial downwash distribution arising, e.g., from incidence, camber or twist [the additional downwash wj g of Eq. 1-2]. In this example, Wkk = 1.0 and fe = FA2J = 0.0 for all of the wing and canard aerodynamic boxes, and wj g = W2GJ = 0.1 deg = 0.001745 rad for the wing boxes and W2GJ = 0.0 for the canard boxes. The SPLINE2 1501 and SET1 1000 entries specify a linear spline on the canard to interconnect the structural and aerodynamic grids. The Cartesian coordinate system CORD2R 1 is for the spline on the canard through the one-quarter chord hinge line. The simplicity of the canard splining makes it uncharacteristic of what can be expected in practice, but it also provides an illustrative example of the use of the smoothing factors DTHX and DTHY. It is seen that DTHX has been set to 1.0, implying that the canard is restrained in roll about the centerline by a rotational spring. If DTHX were set to 0.0, the spline would be overdetermined since the two grid points that lie on the same line perpendicular to the spline axis each have a value for the bending slope when only one is allowed. On the other hand, if DTHX were set to -1.0, the splining would become singular because displacements at two points are not sufficient to define a plane, i.e., rotation of the canard about the centerline would not be precluded. Setting DTHY to -1.0 ensures that only the transverse displacements of the two grid points are used in determining the displacement and rotation of the spline. The SPLINE2 1601 and SET1 1100 entries specify a linear spline on the wing. SET1 1100 includes GRIDs 99 and 100 for a good spline fit to the wing in the root region. CORD2R 2 is the Cartesian coordinate system through the wing elastic axis. CORD2R 100 is for the rigid body motions of the aerodynamic reference point; specifically the pitch and moment axis is at the canard midchord at GRID 90. This coordinate system is the standard NACA body axis system with the x-axis forward and the z-axis downward. The stability derivatives are output using this coordinate system.

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Static Aeroelastic Input The foregoing input is typical for any aeroelastic analysis. The entries in the Bulk Data specifically for static aeroelastic analysis begin with the AESTAT entries, which specify the trim ; parameters. The parameters are angle of attack, α = ANGLEA ; pitch rate, ; and pitch acceleration, . Next, the trim surface normal load factor is defined by an AESURF entry as the elevator (canard) ELEV using coordinate system CORD2R 1 for its hinge line and defining the aerodynamic boxes using AELlST 1000, which specifies aerodynamic box numbers 1000 through 1007. The reference geometry is specified on the AEROS entry. This entry specifies the aerodynamic coordinate system CORD2R 1, the aerodynamic reference coordinate system for rigid body motions CORD2R 100, a reference chord of REFC = = 10.0 ft, a reference span of REFB = b = 40.0 ft (the full span), a reference area of REFS = S = 200.0 sq ft (half-model), and symmetric aerodynamic loading (SYMXZ = 1). The first two TRIM entries specify the flight condition at Mach number, m = 0.9 and level , a one-g load factor, , and flight with no pitch rate, no pitching acceleration, . The first entry, TRIM 1, specifies the low speed condition with dynamic pressure = Q = 40 psf, and the second entry, TRIM 2, specifies = Q = 1200 psf, both at m = 0.9 at sea level. The third entry, TRIM 3, specifies supersonic level flight at m = 1.3 at 20,000 ft with Q = 1151 psf.

Case Control Commands The Case Control Section begins with three title commands. ECHO = BOTH echoes the input data in both unsorted/annotated and in sorted/unannotated formats. SPC = 1 provides the set of symmetric constraints. DISP, STRESS, and FORCE = ALL print all of the displacements, stresses, and forces, respectively. AEROF, APRES = ALL call for all of the aerodynamic forces and pressures to be printed in all subcases: SUBCASE 1 (TRIM = 1) is the low case of level flight at Mach number m = 0.9; SUBCASE 2 (TRIM = 2) is the high case at m = 0.9; SUBCASE 3 is the supersonic flight case at m = 1.3. The OUTPUT(PLOT) request permits graphs to be obtained from auxiliary plotting equipment. In this case, the planform is plotted. In the Executive Control statement, ID NXN, HA144A indicates the identification of this problem. TIME 5 limits CPU time to 5.0 minutes. SOL 144 calls for the Static Aeroelastic Response DMAP sequence, No. 144.

Output The input data for this example is shown in Listing 6-1 followed by the sorted Bulk Data entries in Listing 6-2, and finally the output results in Listing 6-3. The significant output data are discussed and summarized below. The OUTPUT FROM THE GRID POINT WEIGHT GENERATOR gives a centroid of the half-airplane 2.182 ft aft of GRID 90 and 2.500 ft outboard from the centerline, and a centroidal moment of inertia in pitch of 892,894 lb-ft2.

Stability Derivatives The typical definition of the stability derivatives in the restrained longitudinal case may be illustrated by the lift coefficient [see, e.g., Rodden and Love (1985), and note that Cz = −CL ]:

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Equation 6-1. The -derivatives are not obtained from the quasi-steady considerations here and will not be discussed further. The rotations of the mean axis in the restrained Iongitudinal case are defined in terms of rotational derivatives defined by Rodden and Love (1985).

Equation 6-2. Again, the -term will not be discussed further. In the unrestrained case, all inertial derivatives vanish because their effects appear in the remaining derivatives, and the mean axis rotations do not have to be considered in the equations of motion. Four sets of stability derivatives are generated for the system for each flight condition: 1. Rigid unsplined 2. Rigid splined 3. Elastic restrained at the SUPORTed degrees of freedom 4. Elastic unrestrained Before the stability derivatives are tabulated, the transformation from the basic to the reference coordinates is shown. This transformation provides a check on the input of the aerodynamic reference coordinate system for the stability derivatives. The stability derivatives for the rigid and elastic vehicle are shown next. The rigid derivatives are those that are obtained while neglecting elastic deformation of the vehicle. These derivatives are presented in two ways: unsplined and splined coefficients, which provide checks on the splining. The unsplined coefficients are based on all of the boxes in the aerodynamic model and are independent of the spline. Usually, the two sets of coefficients are nearly identical unless there is an error in the spline input, such as not including all of the boxes. However, there may be situations where some boxes intentionally may not be connected to the spline, as in the case when no motion of certain boxes is desired. The stability derivatives are summarized in Table 6-1. For the first dynamic pressure, = Q = 40 psf, the rigid and elastic coefficients are all quite close except that the inertial derivatives have finite values for the low dynamic pressure. In the rigid case, the inertial derivatives vanish, but in the limit of zero dynamic pressure, the flexible inertial derivatives remain finite. By virtue of the definitions of the inertial derivatives in Eq. 6-1, the printed output values corresponding to unit must be divided by . The inertial derivatives are absorbed into the basic stability derivatives in the unrestrained case. The aerodynamic center for each loading may be found by dividing its moment coefficient by its lift coefficient and multiplying by the reference chord, e.g., for the angle of attack α loading in unrestrained flight aft of GRID 90 at low .

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Mean Axis Deformation The mean axis translations and rotations for the SUPORT degrees of freedom follow next in the output and are shown as INTERMEDIATE MATRIX ... HP. The rotational derivatives are presented in the second row (i.e., printed in the second column shown). These derivatives are only associated with the restrained case, since they must be included in the equations of motion that utilize restrained stability derivatives, as discussed by Rodden and Love (1985). Equations of motion relative to the SUPORT using unrestrained stability derivatives are already expressed in terms of mean axis rotations. The mean axis rotational derivatives are also summarized in Table 6-1. The derivative αmα is obtained by adding 1.0 to the tabulated value, and the derivative is obtained by dividing by

.

Table 6-1. Subsonic Derivatives for Example FSW Airplane Unrestrained Restrained Restrained Value for Value at = Value at = Value at = Derivative Rigid Airplane 40 psf 40 psf 1200 psf -0.008421 -0.008464 -0.008509 -0.010332 Czo -0.006008 -0.006031 -0.006064 -0.007074 Cmo -5.071 -5.103 -5.127 -6.463 Czα -2.871 -2.889 -2.907 -3.667 Cmα -0.2461 -0.2538 -0.2520 -0.5430 Czδe 0.5715 0.5667 0.5678 0.3860 Cmδe -12.074 -12.087 -12.158 -12.856 Czq -9.954 -9.956 -10.007 -10.274 Cmq

αmo αmα α m δe αmq

Unrestrained Value at = 1200 psf -0.012653 -0.008678 -7.772 -4.577 -0.5219 0.3956 -16.100 -12.499

0.0

0.003154

-

0.003634

-

0.0

0.002369

-

0.002624

-

0.0

0.01181

-

0.01449

-

0.0

0.007900

-

0.009404

-

-

-0.000004062 0.9980 0.0003336 -0.006942

-

-0.0001419 0.9251 0.006420 -0.2136

-

-

-0.0001624

-

-0.0001108

-

-

-0.001641

-

-0.001457

-

The level flight trim solution follows the mean axis rotations. For the low speed condition, the angle of attack of the structural axis is necessarily high, ANGLEA = α = 0.169191 rad = 9.69 deg, and the corresponding canard incidence is ELEV = δe = 0.492457 rad = 28.22 deg. The aerodynamic pressure and load data follow the trim solution. The pressure coefficients on each box are also high at the high angle of attack and low dynamic pressure. The pressures and box normal forces balance the weight of the airplane. The aerodynamic moments are taken about

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the midchord of each box: at subsonic speeds the box force acts at the box quarter-chord and causes a moment about the box reference midchord; at supersonic speeds the box force acts at the box midchord and results in a zero moment on each box. The restrained and unrestrained derivatives for the second dynamic pressure, = 1200 psf, are shown next and are also summarized in Table 6-1. In this case, the aerodynamic center in unrestrained flight has moved slightly aft to xa.c. = 2.945 ft behind GRID 90. The level flight trim solution is given next. The angle of attack of the structural axis through GRID 90 is α = ANGLEA = 0.001373 rad = 0.079 deg and δe = ELEV = 0.01932 rad = 1.107 deg. Following the trim solution are the aerodynamic pressure coefficients and pressures on each aerodynamic box in the trimmed condition and then the aerodynamic forces and moments about the 50% chord of each box. The aeroelastic redistribution of loads with increasing dynamic pressure is seen by comparing the pressures and normal forces between Subcases 1 and 2. The third subcase is supersonic level flight at m = 1.3 and = 1151 psf. The supersonic stability derivatives are output next and are summarized in Table 6-2. The aerodynamic center has moved aft to 4.088 ft behind GRID 90. Because of the wing incidence of 0.1 deg, the supersonic angle of attack of the structural axis is negative, α = ANGLEA = 0.0005118 rad = −0.003 deg, and the canard angle is δe = ELEV = 0.03027 rad = 1.734 deg. The supersonic load distribution is quite different from the subsonic distribution as can be seen by comparing the pressures and normal forces between Subcases 2 and 3. Next are the structural deformations of all the grid points relative to the SUPORT point, GRID 90, for the three subcases. Finally, the element loads and stresses are given for the three subcases. The loads include bending moments, shears, and torques. Some of the stresses are high for the arbitrarily chosen cross-sectional properties. Table 6-2. Supersonic Derivatives for Example FSW Airplane Restrained Value at Value for Rigid Airplane Derivative = 1151 psf -0.007352 -0.007172 Czo -0.007195 -0.006959 Cmo -4.847 -4.948 Czα -3.885 -3.933 Cmα -0.6346 -0.8386 Czδe 0.2378 0.05436 Cmδe Czq -9.055 -7.611 Cmq -10.149 -8.769

Unrestrained Value at = 1151 psf -0.008557 -0.008270 -5.783 -4.728 -0.8802 0.010545 -9.305 -10.360

0.0

0.002447

-

0.0

0.002470

-

0.0

0.010366

-

0.0

0.009444

-

-

-0.0001459 0.9165 0.00009123

-

αmo αm α αm δ e

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Table 6-2. Supersonic Derivatives for Example FSW Airplane Restrained Value at Value for Rigid Airplane Derivative = 1151 psf αmq -0.1875

Unrestrained Value at = 1151 psf -

-

-0.0001120

-

-

-0.001448

-

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N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA144A $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA144A $$$$$$$$ $ $ $ MODEL DESCRIPTION 30 DEG FWD SWEPT WING W/CANARD $ $ BEAM MODEL WITH DUMBBELL MASSES $ $ $ $ SOLUTION SYMMETRIC IN-FLIGHT STATIC STABILITY $ $ ANALYSIS USING DOUBLET LATTICE $ $ METHOD AERODYNAMICS AT MACH NO. 0.9 $ $ $ $ OUTPUT PLOTS OF THE STICK MODEL AND AERO $ $ GRID, LISTS OF RESTRAINED AND $ $ UNRESTRAINED SYMMETRIC STATIC $ $ STABILITY DERIVATIVES PLUS THE $ $ AERODYNAMIC FORCES AND PRESSURES $ $ PLUS STRESSES AND DEFLECTIONS FOR $ $ 1G LEVEL FLIGHT. $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AEROCEND EXAMPLE HA144A:30 DEG FWD SWEPT WING WITH CANARD PAGE 2 SYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC SYMMETRIC LOADING C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144A: 30 DEG FWD SWEPT WING WITH CANARD 2 SUBTI = SYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO 3 LABEL = HALF-SPAN MODEL, STATIC SYMMETRIC LOADING 4 ECHO = BOTH 5 SPC = 1 $ SYMMETRIC CONSTRAINTS 6 DISP = ALL $ PRINT ALL DISPLACEMENTS 7 STRESS = ALL $ PRINT ALL STRESSES 8 FORCE = ALL $ PRINT ALL FORCES 9 AEROF = ALL $ PRINT ALL AERODYNAMIC FORCES 10 APRES = ALL $ PRINT ALL AERODYNAMIC PRESSURES 11 SUBCASE 1 12 TRIM = 1 $ 1 G LEVEL FLIGHT (LOW SPEED) 13 SUBCASE 2 14 TRIM = 2 $ 1 G LEVEL FLIGHT (HIGH SUBSONIC SPEED) 15 SUBCASE 3 16 TRIM = 3 $ 1 G LEVEL FLIGH (LOW SUPERSONIC SPEED) 17 OUTPUT(PLOT) 18 PLOTTER = NASTRAN 19 SET 1 = ALL 20 FIND SCALE, ORIGIN 1, SET 1 21 PLOT SET 1 22 PLOT STATIC DEFORMATION 0, ORIGIN 1, SET 1, OUTLINE 23 BEGIN BULK EXAMPLE HA144A: 30 DEG FWD SWEPT WING WITH CANARD PAGE 3 SYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC SYMMETRIC LOADING I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT SECTION ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ * * * STRUCTURAL DATA * * * $ $ $ $ (LB-FT-SEC SYSTEM) $ $ $ $ * * GRID GEOMETRY * * $ $ $ $ GRID 90 - 100 (T3) FUSELAGE POINTS $ $ GRID 110 - 122 (T3) WING POINTS $ $ $ $ * FUSELAGE GRID * $ $ $ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $ $ ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS AND $ $ ITS ASSOCIATED SUPERELEMENT ID. $ $ $ $ ID CP X1 X2 X3 CD PS SEID GRID 90 15. 0. 0. GRID 97 0. 0. 0. GRID 98 10. 0. 0. GRID 99 20. 0. 0. GRID 100 30. 0. 0. $ $ $ * WING GRID * $

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$ $ GRID GRID GRID GRID GRID GRID $ $ $ $ $ $ $ $ $ $ $ $ CBAR CBAR CBAR CBAR $ $ $ $ $ $ $ $ $ $ $

$ ID 111 110 112 121 120 122

CP

X1 X2 24.61325 +5. 27.11325 +5. 29.61325 +5. 18.83975+15. 21.33975+15. 23.83975+15.

X3 0. 0. 0. 0. 0. 0.

CD

PS

SEID

* * STRUCTURAL STIFFNESS PROPERTIES * * * FUSELAGE STRUCTURE * THE CBAR ENTRY DEFINES A SIMPLE BEAM ELEMENT. LISTED ARE ITS PROPERTY ENTRY ID, THE TWO GRID POINTS JOINED BY THE BEAM AND COMPONENTS OF A VECTOR FROM THE FIRST POINT. THIS VECTOR DEFINES THE DIRECTION OF THE STRUCTURAL DEFLECTION OF THE POINT AND ITS POSITIVE SENSE. EID 101 102 100 103

PID 100 100 100 100

GA 97 98 90 99

GB 98 90 99 100

X1,GO 0. 0. 0. 0.

X2 0. 0. 0. 0.

X3 1. 1. 1. 1.

THE PBAR ENTRY DEFINES GEOMETRIC PROPERTIES OF THE BEAM. LISTED ARE ITS ASSOCIATED MATERIAL ENTRY ID, ITS CROSS SECTIONAL AREA, AREA MOMENTS OF INERTIA, TORSIONAL MOMENT OF INERTIA AND NON-STRUCTURAL MASS PER UNIT AREA. THE OPTIONAL CONTINUATION ENTRY CONTAINS STRESS RECOVERY COEFFICIENTS, I.E., Y,Z COORDINATES WHERE STRESSES ARE TO BE COMPUTED. K1 AND K2 ARE AREA FACTORS FOR SHEAR STIFFNESS (DEFAULT IS BLANK; THEN SHEAR STIFFNESS IS INFINITE, I.E., SHEAR FLEXIBILITY IS ZERO. I12 IS THE AREA PRODUCT OF INERTIA.

$

$ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $

$ $ PBAR $ +PB1 $ +PB2 $ $ $ $ CBAR CBAR $ $ $ $ $ $ $ $ RBAR RBAR RBAR RBAR $ $ PBAR $ +PB3 $ +PB4 $ $ $ $ $ $ $ MAT1 $ $ $ $ $ $ $ $ $ $ $ CONM2 CONM2 CONM2 CONM2

PID 100 C1 1.0 K1

MID 1 C2 1.0 K2

A 2.0 D1 1.0 I12 0.0

I1 .173611 D2 -1.0

I2 0.15 E1 -1.0

J 0.5 E2 1.0

NSM +PB1 F1 -1.0

F2 -1.0

$ $ $

* WING STRUCTURE * EID 110 120

PID 101 101

GA 100 110

GB 110 120

X1,GO 0. 0.

X2 0. 0.

X3 1. 1.

THE RBAR ENTRY DEFINES A RIGID BAR. LISTED ARE THE GRID POINTS AT EACH END AND THE DEPENDENT AND INDEPENDENT DOFS AT EACH END. THE NUMBER OF INDEPENDENT DOFS AT THE TWO ENDS MUST EQUAL SIX. BY DEFAULT THOSE NOT DECLARED INDEPENDENT ARE MADE DEPENDENT. EID 111 112 121 122

GA 110 110 120 120

GB 111 112 121 122

CNA 123456 123456 123456 123456

CNB

PID 101 C1 0.5 K1

MID 1 C2 3.0 K2

A 1.5 D1 0.5 I12 0.0

I1 I2 0.173611+2.0 D2 E1 -3.0 -0.5

+PB2

CMA

$ $ $ $ $ $ $

CMB

$ J NSM 0.462963 E2 F1 3.0 -0.5

+PB3 F2 -3.0

THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES. LISTED ARE ITS ID, ITS ELASTIC MODULUS, SHEAR MODULUS, POISSONS RATIO, MASS DENSITY, TEMPERATURE EXPANSION COEFFICIENT, REFERENCE TEMPERATURE AND A STRUCTURAL DAMPING COEFFICIENT. MID 1

E 1.44+9

G 5.40+8

NU

RHO

A

TREF

* FUSELAGE MASSES * THE CONM2 ENTRY DEFINES A CONCENTRATED MASS. LISTED ARE ITS ID, GRID LOCATION, COORDINATE SYSTEM TO LOCATE THE CENTER OF GRAVITY, THE MASS VALUE AND THE LOCATION OF THE CENTER OF GRAVITY RELATIVE TO THE GRID LOCATION. G 97 98 99 100

CID 0 0 0 0

M 1500.0 1500.0 1500.0 1500.0

X1

X2

$ $ $ $ $ $

GE

* * MASS AND INERTIA PROPERTIES * *

EID 97 98 99 100

+PB4

$ $ $ $ $ $ $ $ $ $

X3

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$

6-12

Static Aeroelastic Analysis Sample Problems

$ $ * WING MASSES * $ CONM2 111 111 0 600.0 CONM2 112 112 0 400.0 CONM2 121 121 0 600.0 CONM2 122 122 0 400.0 $ $ * * STRUCTURAL PARAMETERS * * $ $ THE PARAM,GRDPNT,XX ENTRY CAUSES THE GRID POINT WEIGHT $ GENERATOR TO BE EXECUTED USING GRID POINT XX AS THE REF$ ERENCE POINT. THEN THE INERTIA MATRIX, THE TRANSFER MATRIX $ FROM BASIC TO PRINCIPAL AXES AND OTHER PERTINENT INERTIA $ DATA ARE PRINTED. $ PARAM GRDPNT 90 $ THE PARAM,WTMASS,GINV CAUSES ALL THE STRUCTURAL MASSES AND $ $ MASS DENSITIES TO BE MULTIPLIED BY GINV (I.E., BY ONE OVER $ THE ACCELERATION OF GRAVITY). THE DYNAMIC PRESSURE SUPPLIED $ FOR AERODYNAMIC FORCE CALCULATIONS WILL NOT BE MULTIPLIED $ BY GINV. $ PARAM WTMASS .031081 $ $ THE PARAM,AUNITS,GINV PERMITS THE ACCELERATIONS ON THE TRIM $ ENTRY TO BE SPECIFIED IN UNITS OF LOAD FACTOR (I.E., IN G’S) $ PARAM AUNITS .031081 $ $ * * STRUCTURAL CONSTRAINTS * * $ $ THE SPC1 ENTRY CONSTRAINS THE LISTED GRID POINTS IN THE $ SPECIFIED DOF COMPONENTS. $ $ SID C G1 G2 G3 G4 SPC1 1 1246 90 SPC1 1 246 97 98 99 100 $ $ THE SUPORT ENTRY IDENTIFIES A GRID POINT OR A SCALAR POINT $ AND SPECIFIES THE DOF COMPONENTS IN WHICH THE USER DESIRES $ REACTIONS TO BE APPLIED TO PREVENT RIGID BODY MOTION. IT $ THUS INVOKES THE SOLUTION OF THE BALANCE EQUATIONS TO DETER$ MINE THE REACTIONS. IN THE STATIC AEROELASTIC SOLUTION $ THE DOF COMPONENTS MUST BE CONSISTENT WITH THE UNDEFINED $ VARIABLES ON THE TRIM ENTRIES. $ $ ID C SUPORT 90 35 $ $ THE OMIT1 ENTRY IDENTIFIES GRID POINT COMPONENTS TO BE OMITTED $ FROM THE REMAINDER OF THE ANALYSIS. $ $ ID G G OMIT1 4 110 120 $ $ * * * AERODYNAMIC DATA * * * $ $ (LB-FT-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AEROS ENTRY IS UNIQUE TO THE STATIC AEROELASTICITY $ SOLUTION, SOL21. ACSID IDENTIFIES THE AERO COORDINATE $ SYSTEM, RCSID IDENTIFIES THE REFERENCE COORDINATE SYS$ TEM FOR RIGID BODY MOTION. REFC IS THE REFERENCE CHORD. $ REFB IS THE REFERENCE SPAN. REFS IS THE REFERENCE WING $ AREA. SYMXZ AND SYMXY ARE SYMMETRY KEYS. $ $ ACSID RCSID REFC REFB REFS SYMXZ SYMXY AEROS 1 100 10.0 40.0 200.0 1 $ $ THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM $ FLAGGED BY THE AEROS ENTRY. THE ORIGIN IS AT THE CANARD $ QUARTER CHORD. LISTED ARE THE ORIGIN, A POINT ALONG THE $ Z AXIS AND A POINT IN THE X-Z PLANE, ALL IN THE RID $ COORDINATE SYSTEM. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0 12.5 0. 0. 12.5 0. 10. $ C1 C2 C3 +CRD1 20. 0. 0. $ $ THIS CORD2R ENTRY DEFINES THE NACA COORDINATE SYSTEM TO $ WHICH ALL THE STABILITY DERIVATIVES AND TRIM CONDITIONS $ WILL BE REFERENCED. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 -10.0 $ C1 C2 C3

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CRD1 $ $ $ $ $ +CRD100

Static Aeroelastic Analysis Sample Problems

+CRD100 0.0 $

0.0

0.0 $

$ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * BEAM SPLINE FIT ON THE WING * $ $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT$ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR $ ATTACHMENT AND TORSIONAL FLEXIBILITIES. CID IDENTIFIES $ THE CORD2R ENTRY THAT DEFINES THE SPLINE AXIS. DTHX AND $ DTHY ARE ROTATIONAL ATTACHMENT FLEXIBILITIES (-1. SPECIFIES $ NO ATTACHMENT). $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 1601 1100 1100 1131 1100 0. 1. 2 $ DTHX DTHY +SPW -1. -1. $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AERODYNAMICS. $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS, $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ EXTRA POINT IDS. $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 1100 1000 8 4 1 $ ( FWD LEFT POINT ) CHORD ( FWD RIGHT POINT ) CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CAW 25. 0. 0. 10. 13.45299+20. 0. 10. $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1000 $ $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 SET1 1100 99 100 111 112 121 122 $ $ THE CORD2R ENTRY DEFINES THE COORDINATE SYSTEM IN WHICH THE $ BEAM SPLINE EXTENDS ALONG THE WING Y-AXIS. IT LISTS THE $ ORIGIN, A POINT ALONG THE Z-AXIS AND A POINT IN THE X-Z $ PLANE. $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 2 0 30. 0. 0. 30. 0. 10. $ C1 C2 C3 +CRD2 38.66025+5.0 0. $ $ * CONTROL SURFACE DEFINITION * $ $ THE AESURF ENTRY DEFINES AN AERODYNAMIC CONTROL SURFACE. $ LISTED ARE THE ALPHANUMERIC NAME OF THE SURFACE, THE ID $ OF A COORDINATE SYSTEM THAT DEFINES THE HINGE LINE AND $ THE ID OF AN AELIST ENTRY. $ $ ID LABEL CID1 ALID1 CID2 ALID2 AESURF 505 ELEV 1 1000 $ $ THE AELIST ENTRY LISTS AERODYNAMIC BOXES THAT LIE ON THE $ CONTROL SURFACE. $ $ SID E1 E2 E3 ETC AELIST 1000 1000 THRU 1007 $ $

* BEAM SPLINE FIT ON THE CANARD * $ $ $ AGRID PANEL (FIRST & LAST BOX)SGRID SPLINE2 1501 1000 1000 1007 1000 0. 1. +SPC 1. -1. $ $ PANEL PID CP NSPAN NCHORD CAERO1 1000 1000 2 4 $ (FWD LEFT POINT ) CHORD (FWD RIGHT POINT

$ $ $ $ $ $ $ $ $ $ $ $ $ $ +SPW $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CAW

$ $ $ $ $ $ $ $

$ $ $ $ $ +CRD2 $ $ $ $ $ $ $ $ $ $ $ $ $

$ SPLCS 1

+SPC $

)

IGP 1 CHORD

+CAC

NX Nastran Aeroelastic Analysis User’s Guide

6-13

Chapter 6

Static Aeroelastic Analysis Sample Problems

+CAC $ $ SET1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DMI $ DMI $ $ * $ DMI DMI DMI DMI $ $ $ DMI DMI DMI DMI $ $ $ $ $ $ $ $ $ $ $ $ AESTAT AESTAT AESTAT AESTAT $ $

6-14

10.

0.

0.

SGRID 1000

GRID POINTS 98 99

10.

10.

5.

0.

10. $

* * * AERODYNAMIC DATA * * * * * USER SUPPLIED INPUT DATA * * THE DMI ENTRY ACCOMMODATES DIRECT INPUT OF USER SUPPLIED MATRICES OF DATA. LISTED ARE THE NAME OF THE MATRIX, THE FORM OF MATRIX (IN THIS CASE DIAGONAL), THE TYPE OF DATA (IN THIS CASE REAL SINGLE PRECISION), BEING INPUT AND THE TYPE EXPECTED AT OUTPUT (IN THIS CASE TO BE DETERMINED INTERNALLY). M IS THE NUMBER OF ROWS AND N IS THE NUMBER OF COLUMNS. THE DATA IS EXPECTED BY COLUMNS. THE CONTINUATION ENTRY LISTS THE COLUMN NO., THE ROW NO. OF THE FIRST NON-ZERO ELEMENT AND THE FOLLOWING ELEMENTS IN THAT COLUMN. * PRESSURE MODIFIERS (WEIGHTING MATRIX) * NAME WKK NAME WKK

"0" 0 J 1

FORM 3 I1 1

TIN 1 A(I1,J) 1.0

TOUT M 0 80 A(I1+1,J) . . . THRU 80

N 1

INITIAL DOWNWASHES (E.G., DUE TO INCIDENCE,TWIST OR CAMBER) * W2GJ W2GJ W2GJ W2GJ

0 1 2 3

2 9 9 9

1 0 .0017453THRU .0017453THRU .0017453THRU

40

0 1 2 3

2 1 1 1

1 0.0 0.0 0.0

0 THRU THRU THRU

3

$ $ $

40

3

40 40 40

* * * SOLUTION SPECIFICATIONS * * * * * AERODYNAMIC DOFS * * THE AESTAT ENTRY LISTS TRIM VARIABLES USED TO SPECIFY RIGID BODY MOTIONS. THESE AND THE CONTROL SURFACE ROTATIONS MAKE UP THE VARIABLES IN THE EQUATIONS OF MOTION. ID 501 502 503 504

$ $ $

40 40 40

* PRESSURES (E.G., AT ZERO ANGLE OF ATTACK) * FA2J FA2J FA2J FA2J

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $

LABEL ANGLEA PITCH URDD3 URDD5

* * TRIM CONDITIONS * * $ $ $ THE TRIM ENTRY SPECIFIES CONSTRAINTS FOR THE TRIM VARIABLES $ LISTED ON THE AESTAT AND AESURF ENTRIES. LISTED ARE ITS ID, $ THE MACH NUMBER, DYNAMIC PRESSURE AND PAIRS OF TRIM VARI$ ABLES AND THEIR CONSTRAINED VALUES. THOSE THAT ARE NOT $ HELD FIXED MUST BE CONSTRAINED BY REACTION FORCES STIPU$ LATED ON THE SUPORT ENTRY. SEE SECTION 3.5.3 OF THE THEO$ RETICAL MANUAL FOR MORE DETAILS. $ $ TRIM CONDITION 1: 1 G LEVEL FLIGHT AT LOW SPEED $ $ ID MACH Q LABEL1 UX1 LABEL2 UX2 TRIM 1 0.9 40.0 PITCH 0.0 URDD3 -1.0 $ LABEL3 UX3 +TR1 URDD5 0.0 $ * * * $ TRIM CONDITION 2: 1 G LEVEL FLIGHT AT HIGH SUBSONIC SPEED $ TRIM 2 0.9 1200.0 PITCH 0.0 URDD3 -1.0 +TR2 URDD5 0.0 $ * * * $ TRIM CONDITION 3: 1 G LEVEL FLIGHT AT LOW SUPERSONIC SPEED $ TRIM 3 1.3 1151.0 PITCH 0.0 URDD3 -1.0 +TR3 URDD5 0.0 $ * * * ENDDATA INPUT BULK DATA CARD COUNT = 388

NX Nastran Aeroelastic Analysis User’s Guide

$

$ $ $ $ $ $ $ $ $ $ $ +TRM +TR1 $ $ $ +TR2 $ $ $ +TR3 $

Static Aeroelastic Analysis Sample Problems

Listing 6-1. Input Files for FSW Airplane in Level Flight

NX Nastran Aeroelastic Analysis User’s Guide

6-15

Static Aeroelastic Analysis Sample Problems

Chapter 6

EXAMPLE HA144A: 30 DEG FWD SWEPT WING WITH CANARD SYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC SYMMETRIC LOADING S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 1AELIST 1000 1000 THRU 1007 2AEROS 1 100 10.0 40.0 200.0 1 3AESTAT 501 ANGLEA 4AESTAT 502 PITCH 5AESTAT 503 URDD3 6AESTAT 504 URDD5 7AESURF 505 ELEV 1 1000 8CAERO1 1000 1000 2 4 9+CAC 10. 0. 0. 10. 10. 5. 0. 10CAERO1 1100 1000 8 4 11+CAW 25. 0. 0. 10. 13.45299+20. 0. 12CBAR 100 100 90 99 0. 0. 1. 13CBAR 101 100 97 98 0. 0. 1. 14CBAR 102 100 98 90 0. 0. 1. 15CBAR 103 100 99 100 0. 0. 1. 16CBAR 110 101 100 110 0. 0. 1. 17CBAR 120 101 110 120 0. 0. 1. 18CONM2 97 97 0 1500.0 19CONM2 98 98 0 1500.0 20CONM2 99 99 0 1500.0 21CONM2 100 100 0 1500.0 22CONM2 111 111 0 600.0 23CONM2 112 112 0 400.0 24CONM2 121 121 0 600.0 25CONM2 122 122 0 400.0 26CORD2R 1 0 12.5 0. 0. 12.5 0. 27+CRD1 20. 0. 0. 28CORD2R 2 0 30. 0. 0. 30. 0. 29+CRD2 38.66025+5.0 0. 30CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 31+CRD100 0.0 0.0 0.0 32DMI FA2J 0 2 1 0 40 33DMI FA2J 1 1 0.0 THRU 40 34DMI FA2J 2 1 0.0 THRU 40 35DMI FA2J 3 1 0.0 THRU 40 36DMI W2GJ 0 2 1 0 40 37DMI W2GJ 1 9 .0017453THRU 40 38DMI W2GJ 2 9 .0017453THRU 40 39DMI W2GJ 3 9 .0017453THRU 40 40DMI WKK 0 3 1 0 80 41DMI WKK 1 1 1.0 THRU 80 42GRID 90 15. 0. 0. 43GRID 97 0. 0. 0. 44GRID 98 10. 0. 0. 45GRID 99 20. 0. 0. 46GRID 100 30. 0. 0. 47GRID 110 27.11325+5. 0. 48GRID 111 24.61325+5. 0. 49GRID 112 29.61325+5. 0. 50GRID 120 21.33975+15. 0. 51GRID 121 18.83975+15. 0. 52GRID 122 23.83975+15. 0. 53MAT1 1 1.44+9 5.40+8 54OMIT1 4 110 120 55PAERO1 1000 56PARAM AUNITS .031081 57PARAM GRDPNT 90 58PARAM WTMASS .031081 59PBAR 100 1 2.0 .173611 0.15 0.5 606162636465666768697071727374757677787980818283-

6-16

+PB1 1.0 +PB2 PBAR 101 +PB3 0.5 +PB4 RBAR 111 RBAR 112 RBAR 121 RBAR 122 SET1 1000 SET1 1100 SPC1 1 SPC1 1 SPLINE2 1501 +SPC 1. SPLINE2 1601 +SPW -1. SUPORT 90 TRIM 1 +TR1 URDD5 TRIM 2 +TR2 URDD5 TRIM 3 +TR3 URDD5 ENDDATA

1.0 1 3.0 110 110 120 120 98 99 246 1246 1000 -1. 1100 -1. 35 0.9 0.0 0.9 0.0 1.3 0.0

NX Nastran Aeroelastic Analysis User’s Guide

1.0 0.0 1.5 0.5 0.0 111 112 121 122 99 100 97 90 1000

-1.0

-1.0

0.173611+2.0 -3.0 -0.5

1.0

-1.0

0.462963 3.0 -0.5

PAGE

..

9

1 10. 1 10.

..

11

10

+CAC +CAW

10.

+CRD1

10.

+CRD2

-10.0

+CRD100

3

3

1

+PB1 -1.0

+PB2

-3.0

+PB3 +PB4

123456 123456 123456 123456 111 98

112 99

121 100

122

1007

1000

0.

1.

1

+SPC

1100

1131

1100

0.

1.

2

+SPW

40.0

PITCH

0.0

URDD3

-1.0

+TR1

1200.0

PITCH

0.0

URDD3

-1.0

+TR2

1151.0

PITCH

0.0

URDD3

-1.0

+TR3

.

Static Aeroelastic Analysis Sample Problems

TOTAL COUNT=

84

Listing 6-2. Sorted Bulk Data Entries for FSW Airplane in Level Flight

NX Nastran Aeroelastic Analysis User’s Guide

6-17

Chapter 6

Static Aeroelastic Analysis Sample Problems

EXAMPLE HA144A: 30 DEG FWD SWEPT WING WITH CANARD PAGE 15 SYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC SYMMETRIC LOADING O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 90 M O * 8.000000E+03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.000000E+04 * * 0.000000E+00 8.000000E+03 0.000000E+00 0.000000E+00 0.000000E+00 1.745300E+04 * * 0.000000E+00 0.000000E+00 8.000000E+03 2.000000E+04 -1.745300E+04 0.000000E+00 * * 0.000000E+00 0.000000E+00 2.000000E+04 2.500000E+05 -1.456625E+05 0.000000E+00 * * 0.000000E+00 0.000000E+00 -1.745300E+04 -1.456625E+05 9.309703E+05 0.000000E+00 * * -2.000000E+04 1.745300E+04 0.000000E+00 0.000000E+00 0.000000E+00 1.180970E+06 * S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 8.000000E+03 0.000000E+00 2.500000E+00 0.000000E+00 Y 8.000000E+03 2.181625E+00 0.000000E+00 0.000000E+00 Z 8.000000E+03 2.181625E+00 2.500000E+00 0.000000E+00 I(S) * 2.000000E+05 1.020300E+05 0.000000E+00 * * 1.020300E+05 8.928944E+05 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.092894E+06 * I(Q) * 9.076062E+05 * * 1.852883E+05 * * 1.092894E+06 * Q * 1.427144E-01 9.897639E-01 0.000000E+00 * * -9.897639E-01 1.427144E-01 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 1 N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T D E R I V A T I V E C O E F F I C I E N T S MACH = 9.0000E-01 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS TRIM VARIABLE COEFFICIENT RIGID INTERCEPT CX 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 CZ -8.420787E-03 -8.420787E-03 CMX 0.000000E+00 0.000000E+00 CMY -6.008128E-03 -6.008129E-03 CMZ 0.000000E+00 0.000000E+00 ANGLEA CX 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 CZ -5.070976E+00 -5.070976E+00 CMX 0.000000E+00 0.000000E+00 CMY -2.870932E+00 -2.870932E+00 CMZ 0.000000E+00 0.000000E+00 PITCH CX 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 CZ -1.207429E+01 -1.207429E+01 CMX 0.000000E+00 0.000000E+00 CMY -9.953999E+00 -9.953999E+00 CMZ 0.000000E+00 0.000000E+00 URDD3 CX 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 URDD5

ELEV

CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ

0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.461395E-01 0.000000E+00 5.715300E-01 0.000000E+00

0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.461395E-01 0.000000E+00 5.715300E-01 0.000000E+00

R O L Q = 4.0000E+01 { 1.5000E+01 } { 0.0000E+00 } { 0.0000E+00 } ELASTIC 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -8.463759E-03 -8.508835E-03 0.000000E+00 0.000000E+00 -6.031396E-03 -6.063685E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -5.103214E+00 -5.127251E+00 0.000000E+00 0.000000E+00 -2.889138E+00 -2.906503E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.208659E+01 -1.215826E+01 0.000000E+00 0.000000E+00 -9.956323E+00 -1.000715E+01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.154314E-03 0.000000E+00 0.000000E+00 0.000000E+00 2.368632E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 5.904781E-02 0.000000E+00 3.950050E-02 0.000000E+00 0.000000E+00 0.000000E+00 -2.537565E-01 0.000000E+00 5.666561E-01 0.000000E+00

INTERMEDIATE MATRIX ... HP0 COLUMN 1 6.451200E-05 -4.062406E-06 INTERMEDIATE MATRIX ... HP COLUMN 1 3.659248E-02 -1.994000E-03 COLUMN 2 9.315132E-02 -6.941710E-03 COLUMN 3 6.349168E-03 -1.624086E-04

6-18

NX Nastran Aeroelastic Analysis User’s Guide

0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.519655E-01 0.000000E+00 5.677909E-01 0.000000E+00

Static Aeroelastic Analysis Sample Problems

COLUMN 1.459012E-02 COLUMN -3.707928E-04

4 -8.205780E-03 5 3.336261E-04

A E R O S T A T I C

AERODYNAMIC GRID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 1.691910E-01 PITCH 0.000000E+00 URDD3 -1.000000E+00 URDD5 0.000000E+00 ELEV 4.924567E-01 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 4.000000E+01 AERODYNAMIC FORCES AERODYNAMIC PRES. AERODYNAMIC LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) LS 4.784522E+00 1.913809E+02 1000 LS 1.196130E+03 7.475815E+02 LS 8.703244E-01 3.481298E+01 1001 LS 2.175811E+02 1.359882E+02 LS 3.553296E-01 1.421318E+01 1002 LS 8.883241E+01 5.552025E+01 LS 2.838606E-01 1.135443E+01 1003 LS 7.096516E+01 4.435323E+01 LS 3.788153E+00 1.515261E+02 1004 LS 9.470382E+02 5.918989E+02 LS 6.253395E-01 2.501358E+01 1005 LS 1.563349E+02 9.770930E+01 LS 2.709481E-01 1.083792E+01 1006 LS 6.773703E+01 4.233564E+01 LS 2.500265E-01 1.000106E+01 1007 LS 6.250663E+01 3.906664E+01 LS -1.200069E+00 -4.800277E+01 1100 LS -3.000173E+02 -1.875108E+02 LS -5.319461E-02 -2.127784E+00 1101 LS -1.329865E+01 -8.311658E+00 LS 3.286648E-02 1.314659E+00 1102 LS 8.216619E+00 5.135388E+00 LS 2.254038E-02 9.016151E-01 1103 LS 5.635094E+00 3.521934E+00 LS -7.478276E-03 -2.991310E-01 1104 LS -1.869569E+00 -1.168481E+00 LS -1.701564E-03 -6.806257E-02 1105 LS -4.253911E-01 -2.658694E-01 LS 4.540962E-02 1.816385E+00 1106 LS 1.135240E+01 7.095254E+00 LS 3.006115E-02 1.202446E+00 1107 LS 7.515286E+00 4.697053E+00 LS 4.197103E+00 1.678841E+02 1108 LS 1.049276E+03 6.557972E+02 LS 4.496382E-01 1.798553E+01 1109 LS 1.124095E+02 7.025597E+01 LS 1.206774E-01 4.827095E+00 1110 LS 3.016934E+01 1.885584E+01 LS 4.692905E-02 1.877162E+00 1111 LS 1.173226E+01 7.332664E+00 LS 3.132609E+00 1.253044E+02 1112 LS 7.831523E+02 4.894702E+02 LS 1.012216E+00 4.048864E+01 1113 LS 2.530540E+02 1.581587E+02 LS 2.907213E-01 1.162885E+01 1114 LS 7.268033E+01 4.542521E+01 LS 9.219103E-02 3.687641E+00 1115 LS 2.304776E+01 1.440485E+01 LS 2.432680E+00 9.730718E+01 1116 LS 6.081699E+02 3.801062E+02 LS 1.065420E+00 4.261680E+01 1117 LS 2.663550E+02 1.664719E+02 LS 4.645951E-01 1.858380E+01 1118 LS 1.161488E+02 7.259298E+01 LS 1.603585E-01 6.414340E+00 1119 LS 4.008963E+01 2.505602E+01 LS 1.921810E+00 7.687241E+01 1120 LS 4.804525E+02 3.002828E+02

30

LS 9.217110E-01 3.686844E+01 1121 LS 2.304277E+02 1.440173E+02 31 LS 5.207017E-01 2.082807E+01 1122 LS 1.301754E+02 8.135963E+01 32 LS 2.245564E-01 8.982255E+00 1123 LS 5.613909E+01 3.508693E+01 33 LS 1.467171E+00 5.868684E+01 1124 LS 3.667927E+02 2.292455E+02 34 LS 7.182633E-01 2.873053E+01 1125 LS 1.795658E+02 1.122287E+02 35 LS 4.659271E-01 1.863708E+01 1126 LS 1.164818E+02 7.280112E+01 36 LS 2.473706E-01 9.894822E+00 1127 LS 6.184264E+01 3.865165E+01 37 LS 9.616055E-01 3.846422E+01 1128 LS 2.404014E+02 1.502509E+02 38 LS 4.610369E-01 1.844147E+01 1129 LS 1.152592E+02 7.203701E+01 39 LS 3.229629E-01 1.291852E+01 1130 LS 8.074072E+01 5.046295E+01 40 LS 2.048080E-01 8.192319E+00 1131 LS 5.120200E+01 3.200125E+01 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 2 N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 9.0000E-01 Q = 1.2000E+03 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { 1.5000E+01 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -8.420787E-03 -8.420787E-03 -1.033208E-02 -1.265279E-02 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -6.008128E-03 -6.008128E-03 -7.073892E-03 -8.678378E-03 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ANGLEA CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -5.070976E+00 -5.070976E+00 -6.462928E+00 -7.771568E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -2.870931E+00 -2.870932E+00 -3.667061E+00 -4.576804E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 PITCH CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -1.207429E+01 -1.207429E+01 -1.285578E+01 -1.610021E+01 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -9.953999E+00 -9.953999E+00 -1.027400E+01 -1.249935E+01 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 6

URDD3

Static Aeroelastic Analysis Sample Problems

0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.634422E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 2.623634E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 URDD5 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 7.244892E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 4.702159E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ELEV 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.461395E-01 -2.461395E-01 -5.429845E-01 -5.219331E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 5.715299E-01 5.715299E-01 3.860484E-01 3.956231E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 INTERMEDIATE MATRIX ... HP0 COLUMN 1 2.373158E-03 -1.418578E-04 INTERMEDIATE MATRIX ... HP COLUMN 1 1.414911E+00 -7.485947E-02 COLUMN 2 2.982306E+00 -2.136336E-01 COLUMN 3 5.546751E-03 -1.107548E-04 COLUMN 4 -1.318791E-03 -7.283140E-03 COLUMN 5 5.516857E-02 6.420408E-03 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 1.373013E-03 PITCH -1.084202E-19 URDD3 -1.000000E+00 URDD5 0.000000E+00 ELEV 1.932495E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 1.501610E-01 1.801932E+02 1000 LS 1.126208E+03 7.038798E+02 2 LS 2.755904E-02 3.307085E+01 1001 LS 2.066928E+02 1.291830E+02 3 LS 1.161091E-02 1.393309E+01 1002 LS 8.708182E+01 5.442614E+01 4 LS 9.609852E-03 1.153182E+01 1003 LS 7.207389E+01 4.504618E+01 5 LS 1.188870E-01 1.426644E+02 1004 LS 8.916525E+02 5.572828E+02 6 LS 1.982195E-02 2.378634E+01 1005 LS 1.486647E+02 9.291541E+01 7 LS 8.882729E-03 1.065928E+01 1006 LS 6.662047E+01 4.163779E+01 8 LS 8.497586E-03 1.019710E+01 1007 LS 6.373189E+01 3.983243E+01 9 LS -3.733832E-02 -4.480598E+01 1100 LS -2.800374E+02 -1.750234E+02 10 LS -3.894551E-03 -4.673461E+00 1101 LS -2.920913E+01 -1.825571E+01 11 LS -8.759687E-04 -1.051162E+00 1102 LS -6.569765E+00 -4.106104E+00 12 LS -2.026481E-04 -2.431778E-01 1103 LS -1.519861E+00 -9.499131E-01 13 LS 5.280314E-03 6.336376E+00 1104 LS 3.960235E+01 2.475147E+01 14 LS -3.127598E-03 -3.753118E+00 1105 LS -2.345699E+01 -1.466062E+01 15 LS -5.930593E-04 -7.116712E-01 1106 LS -4.447945E+00 -2.779966E+00 16 LS -1.144546E-03 -1.373455E+00 1107 LS -8.584092E+00 -5.365057E+00 17 LS 1.412652E-01 1.695183E+02 1108 LS 1.059489E+03 6.621807E+02 18 LS 1.531677E-02 1.838012E+01 1109 LS 1.148757E+02 7.179734E+01 19 LS 2.150892E-03 2.581070E+00 1110 LS 1.613169E+01 1.008231E+01 20 LS -2.372361E-03 -2.846834E+00 1111 LS -1.779271E+01 -1.112044E+01 21 LS 1.080868E-01 1.297041E+02 1112 LS 8.106509E+02 5.066569E+02 22 LS 3.453917E-02 4.144701E+01 1113 LS 2.590438E+02 1.619024E+02 23 LS 9.403754E-03 1.128450E+01 1114 LS 7.052816E+01 4.408010E+01 24 LS 2.046684E-03 2.456020E+00 1115 LS 1.535013E+01 9.593829E+00 25 LS 8.574631E-02 1.028956E+02 1116 LS 6.430974E+02 4.019358E+02 26 LS 3.670283E-02 4.404339E+01 1117 LS 2.752712E+02 1.720445E+02 27 LS 1.572066E-02 1.886479E+01 1118 LS 1.179050E+02 7.369060E+01 28 LS 5.050937E-03 6.061125E+00 1119 LS 3.788203E+01 2.367627E+01 29 LS 6.889284E-02 8.267141E+01 1120 LS 5.166963E+02 3.229352E+02 30 LS 3.221075E-02 3.865290E+01 1121 LS 2.415806E+02 1.509879E+02 31 LS 1.784603E-02 2.141523E+01 1122 LS 1.338452E+02 8.365324E+01 32 LS 7.484504E-03 8.981404E+00 1123 LS 5.613377E+01 3.508361E+01 33 LS 5.325398E-02 6.390478E+01 1124 LS 3.994048E+02 2.496280E+02 34 LS 2.543124E-02 3.051748E+01 1125 LS 1.907343E+02 1.192089E+02 35 LS 1.617978E-02 1.941574E+01 1126 LS 1.213484E+02 7.584273E+01 36 LS 8.442384E-03 1.013086E+01 1127 LS 6.331788E+01 3.957368E+01 37 LS 3.524245E-02 4.229094E+01 1128 LS 2.643184E+02 1.651990E+02 38 LS 1.649208E-02 1.979049E+01 1129 LS 1.236906E+02 7.730661E+01 39 LS 1.132641E-02 1.359169E+01 1130 LS 8.494805E+01 5.309253E+01 40 LS 7.072826E-03 8.487391E+00 1131 LS 5.304620E+01 3.315387E+01 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT.

6-20

CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ

NX Nastran Aeroelastic Analysis User’s Guide

Static Aeroelastic Analysis Sample Problems

HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 3 N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 1.3000E+00 Q = 1.1510E+03 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { 1.5000E+01 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -7.352388E-03 -7.352388E-03 -7.171839E-03 -8.557475E-03 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -7.194669E-03 -7.194669E-03 -6.958829E-03 -8.269967E-03 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ANGLEA CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -4.847300E+00 -4.847300E+00 -4.947799E+00 -5.783336E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -3.884555E+00 -3.884555E+00 -3.932819E+00 -4.727877E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 PITCH CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -9.055174E+00 -9.055174E+00 -7.611267E+00 -9.304661E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -1.014905E+01 -1.014905E+01 -8.768766E+00 -1.036005E+01 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 URDD3 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 2.447211E-03 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 2.469906E-03 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 URDD5 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 5.183127E-02 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 4.722246E-02 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ELEV CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -6.346213E-01 -6.346213E-01 -8.385689E-01 -8.801823E-01 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 2.377555E-01 2.377555E-01 5.436292E-02 1.054454E-02 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 INTERMEDIATE MATRIX ... HP0 COLUMN 1 1.785767E-03 -1.459055E-04 INTERMEDIATE MATRIX ... HP COLUMN 1 1.131482E+00 -8.350787E-02 COLUMN 2 1.929364E+00 -1.875341E-01 COLUMN 3 5.772970E-03 -1.119902E-04 COLUMN 4 2.537520E-03 -7.237968E-03 COLUMN 5 1.082953E-01 9.122791E-05 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA -5.118454E-05 PITCH 0.000000E+00 URDD3 -1.000000E+00 URDD5 0.000000E+00 ELEV 3.027371E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 1.300000E+00 Q = 1.151000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 1.390318E-01 1.600257E+02 1000 LS 1.000160E+03 0.000000E+00 2 LS 1.039268E-01 1.196197E+02 1001 LS 7.476234E+02 0.000000E+00 3 LS 4.975258E-02 5.726522E+01 1002 LS 3.579077E+02 0.000000E+00 4 LS 1.373659E-02 1.581081E+01 1003 LS 9.881757E+01 0.000000E+00 5 LS 1.171181E-01 1.348029E+02 1004 LS 8.425181E+02 0.000000E+00 6 LS 6.698383E-02 7.709839E+01 1005 LS 4.818649E+02 0.000000E+00 7 LS 4.058219E-02 4.671011E+01 1006 LS 2.919382E+02 0.000000E+00 8 LS 1.227154E-02 1.412454E+01 1007 LS 8.827840E+01 0.000000E+00 9 LS -1.002807E-01 -1.154231E+02 1100 LS -7.213945E+02 0.000000E+00 10 LS -3.034289E-02 -3.492467E+01 1101 LS -2.182792E+02 0.000000E+00 11 LS -7.351310E-03 -8.461357E+00 1102 LS -5.288349E+01 0.000000E+00 12 LS -2.369489E-03 -2.727282E+00 1103 LS -1.704551E+01 0.000000E+00 13 LS -2.004906E-02 -2.307646E+01 1104 LS -1.442279E+02 0.000000E+00 14 LS -2.124946E-02 -2.445813E+01 1105 LS -1.528633E+02 0.000000E+00 15 LS -2.301375E-02 -2.648883E+01 1106 LS -1.655552E+02 0.000000E+00 16 LS -5.608012E-03 -6.454822E+00 1107 LS -4.034263E+01 0.000000E+00 17 LS 1.605985E-01 1.848489E+02 1108 LS 1.155306E+03 0.000000E+00 18 LS 3.754428E-02 4.321347E+01 1109 LS 2.700842E+02 0.000000E+00 19 LS -3.846410E-03 -4.427218E+00 1110 LS -2.767011E+01 0.000000E+00 20 LS -2.034224E-02 -2.341392E+01 1111 LS -1.463370E+02 0.000000E+00

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Chapter 6

Static Aeroelastic Analysis Sample Problems

21 LS 7.659681E-02 8.816293E+01 1112 LS 5.510183E+02 22 LS 7.658254E-02 8.814650E+01 1113 LS 5.509156E+02 23 LS 3.448335E-02 3.969034E+01 1114 LS 2.480646E+02 24 LS 2.927190E-03 3.369196E+00 1115 LS 2.105748E+01 25 LS 4.240667E-02 4.881008E+01 1116 LS 3.050630E+02 26 LS 4.723879E-02 5.437184E+01 1117 LS 3.398240E+02 27 LS 5.111507E-02 5.883345E+01 1118 LS 3.677090E+02 28 LS 3.402041E-02 3.915749E+01 1119 LS 2.447343E+02 29 LS 2.497791E-02 2.874958E+01 1120 LS 1.796848E+02 30 LS 2.738711E-02 3.152257E+01 1121 LS 1.970161E+02 31 LS 3.429863E-02 3.947772E+01 1122 LS 2.467358E+02 32 LS 3.888891E-02 4.476114E+01 1123 LS 2.797571E+02 33 LS 1.786715E-02 2.056509E+01 1124 LS 1.285318E+02 34 LS 1.355681E-02 1.560389E+01 1125 LS 9.752431E+01 35 LS 1.867045E-02 2.148969E+01 1126 LS 1.343106E+02 36 LS 2.563084E-02 2.950110E+01 1127 LS 1.843819E+02 37 LS 1.257711E-02 1.447626E+01 1128 LS 9.047660E+01 38 LS 7.189183E-03 8.274750E+00 1129 LS 5.171719E+01 39 LS 6.744027E-03 7.762375E+00 1130 LS 4.851484E+01 40 LS 1.182458E-02 1.361009E+01 1131 LS 8.506307E+01 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. HALF-SPAN MODEL, STATIC SYMMETRIC LOADING D I S P L A C E M E N T POINT ID. TYPE T1 T2 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T3 0.0 -6.176795E-03 -8.566984E-04 -9.633963E-04 -1.065960E-02 -2.420829E-03 3.972288E-03 -8.813946E-03 2.554698E-02 3.368622E-02 1.740775E-02

D I S P L A C E M E N T POINT ID. TYPE T1 T2 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T3 0.0 -6.264164E-03 -8.785404E-04 -1.007080E-03 -1.125761E-02 -2.503219E-03 4.286876E-03 -9.293314E-03 2.737436E-02 3.604945E-02 1.869926E-02

D I S P L A C E M E N T POINT ID. TYPE T1 T2 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T3 0.0 -5.984768E-03 -8.086917E-04 -8.673829E-04 -9.000065E-03 -2.312948E-03 2.641242E-03 -7.267138E-03 1.953163E-02 2.512185E-02 1.394141E-02

0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00

SUBCASE 1

V E C T O R R1 0.0 0.0 0.0 0.0 0.0 7.440053E-04 7.440053E-04 7.440053E-04 1.262856E-03 1.262856E-03 1.262856E-03

R2 0.0 -6.320098E-04 -3.320095E-04 3.960283E-04 1.688768E-03 2.557247E-03 2.557247E-03 2.557247E-03 3.255694E-03 3.255694E-03 3.255694E-03

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 SUBCASE 2

R2 0.0 -6.385624E-04 -3.385622E-04 4.156862E-04 1.789539E-03 2.716038E-03 2.716038E-03 2.716038E-03 3.470037E-03 3.470037E-03 3.470037E-03

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 SUBCASE 3

R2 0.0 -6.176077E-04 -3.176075E-04 3.528223E-04 1.363731E-03 1.981676E-03 1.981676E-03 1.981676E-03 2.236088E-03 2.236088E-03 2.236088E-03

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 SUBCASE 1

V E C T O R R1 0.0 0.0 0.0 0.0 0.0 7.962558E-04 7.962558E-04 7.962558E-04 1.358045E-03 1.358045E-03 1.358045E-03 V E C T O R R1 0.0 0.0 0.0 0.0 0.0 6.497573E-04 6.497573E-04 6.497573E-04 1.088871E-03 1.088871E-03 1.088871E-03

F O R C E S I N B A R E L E M E N T S ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 100 -1.820093E+04 0.0 -2.140187E+04 0.0 6.401870E+02 101 -9.094947E-13 0.0 -1.500000E+04 0.0 1.500000E+03 102 -1.500000E+04 0.0 -1.820094E+04 0.0 6.401870E+02 103 -2.140187E+04 0.0 -4.323508E+04 0.0 2.183321E+03 110 5.796102E+04 0.0 3.544555E+04 0.0 3.899794E+03 120 3.026060E+04 0.0 4.318098E+03 0.0 2.246687E+03 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

( C B A R AXIAL PLANE 2 0.0 0.0 0.0 0.0 0.0 0.0

) FORCE TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 45978E+04 0.0 479167E+03 SUBCASE 2

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE 100 -1.885620E+04 0.0 -2.271239E+04 0.0 7.712394E+02 0.0 0.0 101 9.094947E-13 0.0 -1.500000E+04 0.0 1.500000E+03 0.0 0.0 102 -1.500000E+04 0.0 -1.885620E+04 0.0 7.712394E+02 0.0 0.0

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TORQUE 0.0 0.0 0.0

Static Aeroelastic Analysis Sample Problems

103 -2.271239E+04 0.0 -4.598023E+04 110 6.164220E+04 0.0 3.819550E+04 120 3.274051E+04 0.0 4.651097E+03 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

0.0 0.0 0.0

2.326784E+03 4.061087E+03 2.432615E+03

0.0 0.0 0.0

0.0 0.0 0.0

0.0 1.750427E+04 8.055939E+03 SUBCASE 3

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 100 -1.676073E+04 0.0 -1.852147E+04 0.0 3.521470E+02 0.0 0.0 0.0 101 0.0 0.0 -1.500000E+04 0.0 1.500000E+03 0.0 0.0 0.0 102 -1.500000E+04 0.0 -1.676073E+04 0.0 3.521470E+02 0.0 0.0 0.0 103 -1.852147E+04 0.0 -3.202391E+04 0.0 1.350244E+03 0.0 0.0 0.0 110 4.827697E+04 0.0 2.721261E+04 0.0 3.648454E+03 0.0 0.0 9.105316E+03 120 2.196529E+04 0.0 9.632417E+00 0.0 1.901416E+03 0.0 0.0 1.668437E+01 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 1 S T R E S S E S I N B A R E L E M E N T S ( C B A R ) ELEMENT SA1 SA2 SA3 SA4 AXIAL SA-MAX SA-MIN M.S.-T ID. SB1 SB2 SB3 SB4 STRESS SB-MAX SB-MIN M.S.-C 100 1.048374E+05 1.048374E+05 -1.048374E+05 -1.048374E+05 0.0 1.048374E+05 -1.048374E+05 1.232748E+05 1.232748E+05 -1.232748E+05 -1.232748E+05 1.232748E+05 -1.232748E+05 101 5.238693E-12 5.238693E-12 -5.238693E-12 -5.238693E-12 0.0 5.238693E-12 -5.238693E-12 8.640005E+04 8.640005E+04 -8.640005E+04 -8.640005E+04 8.640005E+04 -8.640005E+04 102 8.640005E+04 8.640005E+04 -8.640005E+04 -8.640005E+04 0.0 8.640005E+04 -8.640005E+04 1.048375E+05 1.048375E+05 -1.048375E+05 -1.048375E+05 1.048375E+05 -1.048375E+05 103 1.232748E+05 1.232748E+05 -1.232748E+05 -1.232748E+05 0.0 1.232748E+05 -1.232748E+05 2.490342E+05 2.490342E+05 -2.490342E+05 -2.490342E+05 2.490342E+05 -2.490342E+05 110 -1.669278E+05 -1.669278E+05 1.669278E+05 1.669278E+05 0.0 1.669278E+05 -1.669278E+05 -1.020832E+05 -1.020832E+05 1.020832E+05 1.020832E+05 1.020832E+05 -1.020832E+05 120 -8.715058E+04 -8.715058E+04 8.715058E+04 8.715058E+04 0.0 8.715058E+04 -8.715058E+04 -1.243613E+04 -1.243613E+04 1.243613E+04 1.243613E+04 1.243613E+04 -1.243613E+04 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING S T R E S S E S ELEMENT SA1 SA2 SA3 ID. SB1 SB2 SB3 100 1.086118E+05 1.086118E+05 -1.086118E+05 1.308235E+05 1.308235E+05 -1.308235E+05 101 -5.238693E-12 -5.238693E-12 5.238693E-12 8.640005E+04 8.640005E+04 -8.640005E+04 102 8.640005E+04 8.640005E+04 -8.640005E+04 1.086118E+05 1.086118E+05 -1.086118E+05 103 1.308235E+05 1.308235E+05 -1.308235E+05 2.648463E+05 2.648463E+05 -2.648463E+05 110 -1.775296E+05 -1.775296E+05 1.775296E+05 -1.100031E+05 -1.100031E+05 1.100031E+05 120 -9.429273E+04 -9.429273E+04 9.429273E+04 -1.339517E+04 -1.339517E+04 1.339517E+04 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING S T R E S S E S ELEMENT SA1 SA2 SA3 ID. SB1 SB2 SB3 100 9.654189E+04 9.654189E+04 -9.654189E+04 1.066837E+05 1.066837E+05 -1.066837E+05 101 0.0 0.0 0.0 8.640005E+04 8.640005E+04 -8.640005E+04 102 8.640005E+04 8.640005E+04 -8.640005E+04 9.654189E+04 9.654189E+04 -9.654189E+04 103 1.066837E+05 1.066837E+05 -1.066837E+05 1.844578E+05 1.844578E+05 -1.844578E+05 110 -1.390378E+05 -1.390378E+05 1.390378E+05 -7.837237E+04 -7.837237E+04 7.837237E+04 120 -6.326009E+04 -6.326009E+04 6.326009E+04 -2.774138E+01 -2.774138E+01 2.774138E+01

SUBCASE 2 B A R E L E M E N T S ( C B A R ) SA4 AXIAL SA-MAX SA-MIN M.S.-T SB4 STRESS SB-MAX SB-MIN M.S.-C -1.086118E+05 0.0 1.086118E+05 -1.086118E+05 -1.308235E+05 1.308235E+05 -1.308235E+05 5.238693E-12 0.0 5.238693E-12 -5.238693E-12 -8.640005E+04 8.640005E+04 -8.640005E+04 -8.640005E+04 0.0 8.640005E+04 -8.640005E+04 -1.086118E+05 1.086118E+05 -1.086118E+05 -1.308235E+05 0.0 1.308235E+05 -1.308235E+05 -2.648463E+05 2.648463E+05 -2.648463E+05 1.775296E+05 0.0 1.775296E+05 -1.775296E+05 1.100031E+05 1.100031E+05 -1.100031E+05 9.429273E+04 0.0 9.429273E+04 -9.429273E+04 1.339517E+04 1.339517E+04 -1.339517E+04 SUBCASE 3 I N B A R E L E M E N T S ( C B A R ) SA4 AXIAL SA-MAX SA-MIN M.S.-T SB4 STRESS SB-MAX SB-MIN M.S.-C -9.654189E+04 0.0 9.654189E+04 -9.654189E+04 -1.066837E+05 1.066837E+05 -1.066837E+05 0.0 0.0 0.0 0.0 -8.640005E+04 8.640005E+04 -8.640005E+04 -8.640005E+04 0.0 8.640005E+04 -8.640005E+04 -9.654189E+04 9.654189E+04 -9.654189E+04 -1.066837E+05 0.0 1.066837E+05 -1.066837E+05 -1.844578E+05 1.844578E+05 -1.844578E+05 1.390378E+05 0.0 1.390378E+05 -1.390378E+05 7.837237E+04 7.837237E+04 -7.837237E+04 6.326009E+04 0.0 6.326009E+04 -6.326009E+04 2.774138E+01 2.774138E+01 -2.774138E+01 I N

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Listing 6-3. Output for FSW Airplane in Level Flight

6.3

Jet Transport Wing in Roll (Example HA144B)

The second example is the BAH wing in steady roll. This is the jet transport wing considered throughout Bisplinghoff, Ashley, and Halfman (1955) and adapted as an demonstration problem by Rodden, Harder, and Bellinger (1979). The present idealization of the wing/aileron combination is shown in Figure 6-2 (note that the figure shows a more refined aerodynamic box idealization than that used previously). The reference geometrical characteristics for the stability derivatives are the wing area, S = 81,250 in2 per side, the reference chord, = 162.5 in., and the total span, b = 1000.0 in. The BAH wing flexibility influence coefficients and mass matrix are given by Rodden (1959a) for the idealization of Figure 6-2. In addition to the data of Rodden (1959a), the half-fuselage is assumed to have a weight of 17,400 lbs with pitch and roll moments of inertia of 4.35 × 109 lb-in2 and 4.37 × 107 lb-in2, respectively. [Note that Bisplinghoff, Ashley, and Halfman (1955, pp. 176-184) give the half-fuselage weight as 17,400 lbs and the half-airplane moment of inertia in roll as 135.02 lb-in2. The moment of inertia in roll of the half-fuselage is derived to be 4.37 × 107 lb-in2, and the moment of inertia in pitch is arbitrarily assumed to be 4.35 × 107 lb-in2] The aileron actuator stiffness is derived from the data in Bisplinghoff, Ashley, and Halfman (1955, pp. 580-584) to give an uncoupled rotational frequency of 60.0 Hz. The wing is idealized into 58 aerodynamic boxes, five of which are on the fuselage and six of which are on the aileron. The wing is assumed to be flying in incompressible air (Mach number, m = 0.0) and at a dynamic pressure = 4.0075 psi for comparison to a Fourier transform solution by Rodden, Harder, and Bellinger (1979). The BAH wing is used in several examples throughout this guide. In the interest of saving space in the guide, certain groups of Bulk Data are contained in separate input files and are incorporated into the input data section by means of the INCLUDE entry. In this example the structural, mass, aileron, and aerodynamic planform data are separated as discussed in the following. The structural stiffness data are input using GENEL entries since the structural model of the BAH wing is given only in terms of flexibility influence coefficients. The GRID entries list the wing grid points 1 through 10, which are the control points for the flexibility matrix, and GRID 11 represents the rigid fuselage at the root of the wing elastic axis. The GENEL partitions include the flexibility matrix Z and the rigid body mode matrix S for plunging, rolling about the centerline, pitching about the elastic axis, and flapping about the wing root. The structural stiffness data are contained in the input file BAH_STRUCT.DAT and are shown in Listing 6-4.

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Figure 6-2. Idealization of BAH Wing for Rolling Analysis

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The wing inertial data are derived from the three masses on each wing strip given by Rodden (1959a) and are input in the CMASS2 format, which provides for the inertial coupling between the forward (25% chord) and aft (75% chord) degrees of freedom. CONM1 is the concentrated rigid fuselage mass, static unbalance, and inertia. PARAM,GRDPNT specifies GRID 11 as the reference in the weight and balance analysis. PARAM,WTMASS = 1/g converts the input weights to units of mass. The inertial data are contained in the input file BAH_MASS.DAT and are shown in Listing 6-5. The relative rotation between the aileron and the wing is represented by GRID 12. GRID 12 is located streamwise behind GRIDs 7 and 8 on the wing trailing edge slightly inboard of the aileron inboard edge. MPC 1 defines the control surface rotation relative to the wing based on the chord lengths between GRIDs 7 and 8 (66.50 in.) and between GRIDs 8 and 12 (33.25 in) at wing station 368. CONM1 2 gives the aileron inertial characteristics: weight (0.0 lbs), unbalance about the hinge line (0.0 in-lbs), and moment of inertia about the hinge line (13,970.lb-in2). CELAS2 3 provides the rotational stiffness of the aileron actuator and is derived from the moment of inertia and the assumed uncoupled frequency of 60.0 Hz [13970.5/g) · (2π · 60.0)2 = 5,142,661 in-lb/rad]. The aileron data are contained in the input file BAH_AILERON.DAT and are shown in Listing 6-6.

Aerodynamic Data The definition of the wing geometry begins with CORD2R 1, which has its x-axis in the streamwise direction. The wing planform is divided into three panels via combinations of three CAERO1 entries and four AEFACT entries. The panel between the centerline and the aileron is specified by CAERO1 1001; by NCHORD = 5, which divides it into five equal chordwise boxes; and by AEFACT 1, which divides it into seven unequal spanwise strips. The panel containing the aileron is specified by CAERO1 2001 and is divided spanwise into three strips and chordwise into six boxes by AEFACTs 2 and 4, respectively. Finally, CAERO1 3001 specifies the wing tip with NCHORD = 5 and AEFACT 3. PAERO1 1000 is the required aerodynamic property entry although no body aerodynamics are considered in this example. The fuselage aerodynamics are found from the extension of the wing between the centerline and the side of the fuselage.

Spline Data Linear splines are used to connect the wing aerodynamics to the wing structure, consisting of grid points listed on SET1 14. SPLlNE2 101 connects the first 35 boxes to the wing structure. Since the GENEL has no rotations, DTHX = DTHY = -1.0. SPLINE2 102 connects boxes 2001 through 2004, 2007 through 2010, and 2013 through 2016 to the wing structure (see Remark 1 on the SPLINE1 Bulk Data entry). SPLINE2 103 connects boxes 3001 through 3005 to the wing structure. The aerodynamic elements are specified in the input file BAH_AERO58.DAT as shown in Listing 6-7. The requirements in the Bulk Data Section for the static aeroelastic analysis begin with the AESTAT entries that specify the antisymmetric trim parameters, roll rate, pb/2V = ROLL, and rolling acceleration, = URDD4. The aileron rotation δa is defined on AESURF as AILE using the hinge line coordinate system CORD2R 10 and by defining the aileron doublet-lattice boxes on AELIST 2005 that lists the six boxes: 2005, 2006, 2011, 2012, 2017 and 2018. SPLINE 104 specifies a plane through the aileron and connects the three SET 15 grid points to the six aileron boxes. The geometry is given on AEROS that specifies CORD2R 1 as both the aerodynamic and rigid body motion reference coordinate systems, a reference chord of REFC = = 162.5 in, a reference span of REFB = b = 1000.0 in, a reference area of REFS = S = 81,250.0 in2, and antisymmetric aerodynamic motion (SYMXZ = -1). The TRIM entry specifies the flight condition at Mach number, m = 0.0, with a dynamic pressure, Q = 4.00075 psi [corresponding to V = 475 mph in Bisplinghoff, Ashley, and Halfman (1955) and the units used in the present analysis], and

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a steady roll maneuver, = URDD4 = 0.0, with unit aileron rotation δa = AILE = 1.0 radian. PARAM,AUNITS"], = 1/g = 0.0025907 permits accelerations to be input in load factor units. ENDDATA completes the Bulk Data Section.

Case Control Commands The Case Control Section begins with three title commands. ECHO = BOTH calls for both annotated and sorted input entries to be listed. SPC = 13 specifies the fuselage (GRID 11) to be constrained in plunge and pitch. MPC = 1 gives the relative angle between the control surface and the wing in terms of wing vertical displacements at GRIDs 7 and 8 and the trailing edge vertical displacement at GRID 12. DlSP = 2 prints the displacements of SET 2, which includes GRIDs in the aileron region of the wing. SPCF = 3 prints the SPC forces on the SET 3 grid points, which is in this case the fuselage at GRID 11. AEROF = ALL and APRES = ALL call for all of the pressures and forces on the aerodynamic boxes to be printed. TRIM 1 specifies the maneuvering condition. BEGIN BULK ends the Case Control Section. In the Executive Control Section, ID NXN, HA144B indicates this test problem identification. TIME 5 specifies the maximum CPU time at 5.0 minutes. SOL 144 calls for the static aeroelastic solution sequence. CEND concludes the Executive Control Section.

Output The input data files are shown in Listing 6-8 followed by the sorted Bulk Data entries in Listing 6-9 and the output in Listing 6-10. The highlights of the computed results are discussed below. The lateral stability derivatives are

Equation 6-3. while the rotation of the lateral mean axis is given by

Equation 6-4. In the unrestrained case, the inertial derivative vanishes, its effect is included in the other two derivatives, and the rotations are not needed in the rolling equation of motion. At the Mach number m = 0.0, the restrained derivatives are found to be

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The output value for URDD4 is ; therefore, a division by b/2 = 500 in. leads to the value above. The rotational derivatives are found from the intermediate matrix HP to be

where is obtained from the output by dividing by b/2. In the unrestrained case, the derivatives are

and

The trim solution gives a rolling helix angle of pb 2V = −Clδa δa/Clp = 0.203284 for the aileron command of δa = 1.0 rad. A dynamic response solution gives pb/2Vδa = 0.197 from its graphical solution (see Example HA146B ). The remaining output of interest gives the pressure coefficients, pressures, and aerodynamic box forces in the steady roll, and the deformations near the wing tip (GRIDs 7, 8, 9, and 10) and at the aileron trailing edge (GRID 12). The rotation of the aileron actuator spring is given by GRID12, R2 and is -0.05056 rad; this is the aeroelastic effect on the aileron, i.e., the net commanded aileron rotation is only 1.0 − 0.05056 = 0.9435 rad. Structural element loads and stresses are not available in this example since the GENEL stiffness model does not contain any details of the structure. The complete solution for aileron effectiveness as a function of dynamic pressure is shown in Figure 6-3. The aileron reversal dynamic pressure is found to be = 11.06 psi by interpolation and corresponds to V = 789 mph at sea level based on the incompressible aerodynamics assumed.

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Note that the curve in Figure 6-3 is not a straight line, although in this example its curvature is very small. Note also that the curve has not included the reduction in commanded aileron rotation due to the actuator flexibility.

Figure 6-3. Aileron Effectiveness of BAH Wing

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Chapter 6

Static Aeroelastic Analysis Sample Problems

$*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ * * * STRUCTURAL DATA * * * $ $ $ $ (LB-IN-SEC SYSTEM) $ $ $ $ * * GRID GEOMETRY * * $ $ $ $ GRID 1 - 10 (T3) WING CONTROL POINTS $ $ GRID 11 (T3,R1,R2) BODY $ $ (R3) WING ROOT HINGE $ $ GRID 12 (T3) AILERON TRAILING EDGE CONTROL POINT $ $ (R2) AILERON RELATIVE ROTATION $ $ $ $ $ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $ $ ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS, AND $ $ ITS ASSOCIATED SUPERELEMENT ID. $ $ $ $ THE BAH JET TRANSPORT WING, AS SHOWN ON P.45 OF THE BOOK $ $ "AEROELASTICITY" BY BISPLINGHOFF, ASHLEY AND HALFMAN, IS $ $ ROTATED 180 DEG AROUND THE Y AXIS. THAT ORIENTATION IS $ $ RETAINED HERE. POINTS 1 THRU 10 ARE ALONG THE ONE- AND $ $ THREE-QUARTER CHORD LINES, POINT 11 IS AT THE ROOT OF THE $ $ ELASTIC AXIS (35% CHORD), $ $ . $ $ ID CP X1 X2 X3 CD PS SEID GRID 1 20.25 90. 12456 GRID 2 -81. 90. 12456 GRID 3 17.85 186. 12456 GRID 4 -71.4 186. 12456 GRID 5 15.8 268. 12456 GRID 6 -63.2 268. 12456 GRID 7 13.3 368. 12456 GRID 8 -53.2 368. 12456 GRID 9 11.05 458. 12456 GRID 10 -44.2 458. 12456 GRID 11 0.0 0. 126 $ * * STRUCTURAL STIFFNESS PROPERTIES * * $ $ $ $ * FLEXIBILITY INFLUENCE COEFFICIENTS * $ $ $ $ THE GENEL ENTRY DEFINES A GENERAL ELEMENT IN TERMS OF ITS $ $ STRUCTURAL INFLUENCE COEFFICIENTS. IT LISTS THE ELEMENT $ $ ID NO. AND PAIRS OF GRID POINT NUMBERS PLUS THEIR UNCON$ $ STRAINED DOFS. THIS IS FOLLOWED BY THE CORRESPONDING PAIRS $ $ THAT WERE CONSTRAINED TO OBTAIN THE INFLUENCE COEFFICIENTS. $ $ THIS IS FOLLOWED BY THE LOWER TRIANGULAR PART OF THE MATRIX $ $ OF INFLUENCE COEFFICIENTS. FINALLY, A MATRIX OF GEOMETRIC $ $ CONSTANTS IS LISTED. THESE CONSTANTS PRODUCE TOTAL FORCES $ $ AND MOMENTS DUE TO DEFLECTIONS IN EACH MODE, IN THIS CASE $ $ LIFT, ROLLING MOMENT, PITCHING MOMENT AND WING ROOT BENDING $ $ MOMENT. $ $ $ $ EID UI1 CI1 UI2 CI2 UI3 CI3 GENEL 432 1 3 2 3 3 3 +01 $ UI4 CI4 UI5 CI5 UI6 CI6 UI7 CI7 +01 4 3 5 3 6 3 7 3 +02 $ UI8 CI8 UI9 CI9 UI10 CI10 +02 8 3 9 3 10 3 +03 $ "UD" UD1 CD1 UD2 CD2 UD3 CD3 +03 UD 11 3 11 4 11 5 +04 $ UD4 CD4 +04 11 6 +05 $ "K"|"Z" Z11 Z21 Z31 ETC (BY COLUMNS) +05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 +06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5 +07 +07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5 +08 +08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5 +09 +09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4 +10 +10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4 +11 +11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4 +12 $ "S" S11 S12 S13 ETC (BY ROWS) +12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 +13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 +14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 +15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 +16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 +17 413.0 $ $

Listing 6-4. BAH_STRUCT.DAT Input File

6-30

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Static Aeroelastic Analysis Sample Problems

$ $ $ $ $ $ $ $ $ $ $ CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 CMASS2 $ $ $ $ $ $ $ $ $ CONM1 $ +51 $ +52 $ $ $ $ $ $ $ $ $ PARAM $ $ $ $ $ $ $ PARAM $

$ $ $ $ $ $ $ $ $ $

* * MASS AND INERTIA PROPERTIES * * * WING MASSES * THE CMASS2 ENTRY DEFINES A SCALAR MASS ELEMENT WITHOUT REFERENCE TO A PROPERTY ENTRY. IT LISTS THE MASS, THE GRID NO. AND ITS DOF COMPONENTS. WHEN TWO GRID POINTS ARE LISTED THE MASS IS ADDED TO BOTH POINTS. EID 121 122 123 341 342 343 561 562 563 781 782 783 9101 9102 9103

M 5248.7 134.9 790.3 9727. 11005. 473. 3253.6 -139.7 946.3 2617.8 21. 782.3 494.8 -7.3 185.2

G1 1 1 2 3 3 4 5 5 6 7 7 8 9 9 10

C1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

G2

C2

2

3

4

3

6

3

8

3

10

3

* FUSELAGE MASS AND INERTIA VALUES * THE CONM1 ENTRY DEFINES A 6 BY 6 SYMMETRIC INERTIA MATRIX FOR A GRID POINT. LISTED IS THE ID, THE GRID POINT NO., THE COORDINATE SYSTEM IN WHICH THE INERTIA MATRIX IS DEFINED AND THE LOWER LEFT TRIANGULAR PART OF THE MATRIX. EID G 1 11 M33 M41 17400. M54 M55 4.35+09

CID

M11

M21

M22

M31

M32

M42

M43

M44 4.37+7 M63

M51

M52

M53

+51 M61

M62

+52 M64

M65

M66

* * STRUCTURAL PARAMETERS * * THE PARAM,WTMASS,GINV CAUSES ALL THE STRUCTURAL MASSES AND MASS DENSITIES TO BE MULTIPLIED BY GINV (I.E., BY ONE OVER THE ACCELERATION OF GRAVITY). THE DYNAMIC PRESSURE SUPPLIED FOR AERODYNAMIC FORCE CALCULATIONS WILL NOT BE MULTIPLIED BY GINV. WTMASS

$ $ $ $ $ $ $ $ $

.0025907

THE PARAM,GRDPNT,XX ENTRY CAUSES THE GRID POINT WEIGHT GENERATOR TO BE EXECUTED USING GRID POINT XX AS THE REFERENCE POINT. THEN THE INERTIA MATRIX, THE TRANSFER MATRIX FROM BASIC TO PRINCIPAL AXES AND OTHER PERTINENT INERTIA DATA ARE PRINTED. GRDPNT

$ $ $ $ $ $ $ $

$ $ $ $ $ $ $

11 $

Listing 6-5. BAH_MASS.DAT Input File

NX Nastran Aeroelastic Analysis User’s Guide

6-31

Static Aeroelastic Analysis Sample Problems

Chapter 6

$ $ $ $ $ GRID $ $ $ $ $ $ $ CELAS2 $ $ $ CONM1 +AIL1 +AIL2 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ MPC $ +MPC1

GRID 12 IS INBOARD OF THE AILERON AND ON THE TRAILING EDGE; IT IS ALIGNED STREAMWISE BEHIND GRIDS 7 AND 8 AND PROVIDES THE MEANS TO INCLUDE THE AILERON IN THE ANALYSIS. ID 12

CP

X1 -86.45

X2 368.

X3

CD

PS 1246

SEID

THE CELAS2 ENTRY DEFINES A SCALAR SPRING ELEMENT WITHOUT REFERENCE TO A PROPERTY ENTRY, IN THIS CASE AN AILERON HINGE SPRING STIFFNESS. IT LISTS THE ID, THE STIFFNESS, THE CONNECTION POINT AND DOF COMPONENT. EID 3

K G1 5142661.12

C1 5 $ $

* * AILERON INERTIAL PROPERTIES * * 2 0.0

$ $ $ $ $ $

12 0.0

+AIL1 +AIL2

13970.5 THE MPC ENTRY DEFINES A MULTIPOINT CONSTRAINT IN THE FORM OF A LINEAR EQUATION. IT LISTS A SET OF TRIPLES CONSISTING OF THE GRID NO., THE CONSTRAINED DOF COMPONENTS AND THE LINEAR COEFFICIENT. THIS ONE SPECIFIES THAT THE Z DISPLACEMENT AT THE TRAILING EDGE OF THE AILERON IS A LINEAR EXTRAPOLATION FROM POINTS 7 AND 8 PLUS THE DISTANCE FROM THE HINGE-LINE TO THE TRAILING EDGE TIMES A UNIT (SMALL), ANGULAR ROTATION OF THE AILERON. SEE P.3.5-9 OF THE "HANDBOOK FOR DYNAMIC ANALYSIS" FOR A DISCUSSION OF THE LAGRANGE MULTIPLIER METHOD WHICH IS USED HERE TO INTRODUCE THE AILERON ROTATION DOF. SID 1 7

G 12 G 3

C 3 C -0.5

A -1.0 A 12

G 8 G 5

C 3 C 33.25

A 1.5 A

$ $ $ $ $ $ $ $ $ $ $ $ $ +MPC1

Listing 6-6. BAH_AILERON.DAT Input File $ THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM $ $ FLAGGED BY THE AEROS ENTRY. THE ORIGIN IS AT THE ROOT $ $ OF THE ELASTIC AXIS. LISTED ARE THE ORIGIN, A POINT $ $ ALONG THE Z AXIS AND A POINT IN THE X-Z PLANE, ALL IN $ $ THE RID COORDINATE SYSTEM. $ $ $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0. 0. 0. 0. .0 -1. +C1 $ C1 C2 C3 +C1 -1. 0. 0. $ $ $ $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AERODYNAMICS. $ $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS, $ $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ $ EXTRA POINT IDS. $ $ $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 1001 1000 0 5 1 1 +CA1 $ ( FWD LEFT POINT ) ROOTCHORD ( FWD RIGHT POINT ) TIP CHORD +CA1 78.75 0. 0. 225. 35. 500. 0. 100. $ CAERO1 2001 1000 0 2 4 1 +CA2 +CA2 78.75 0. 0. 225. 35. 500. 0. 100. $ CAERO1 3001 1000 0 5 3 1 +CA3 +CA3 78.75 0. 0. 225. 35. 500. 0. 100. $ $ $ THE AEFACT ENTRY IS A UTILITY ENTRY USED TO SPECIFY LISTS OF $ $ NUMBERS. IN THIS EXAMLPLE THEY ARE IDENTIFIED BY THE ABOVE $ $ CAERO1 ENTRIES. THE FIRST ENTRY DEFINES THE SPANWISE DIVISIONS $ $ INBOARD OF THE AILERON. $ $ $ $ THE SECOND ONE DEFINES THE SPANWISE DIVISIONS ACROSS THE $ $ AILERON. $ $ $ $ THE THIRD ONE DEFINES THE SPANWISE DIVISIONS OF THE TIP $ $ FAIRING. $ $ $ $ THE FOURTH ONE DEFINES THE CHORDWISE DIVISIONS OF THE $ $ AILERON. THE AILERON HINGE-LINE IS AT THE THREE-QUARTER $ $ CHORD LINE SO THERE ARE TWO CHORDWISE BOXES ON THE $ $ AILERON. $ $ $ NX Nastran Aeroelastic Analysis User’s Guide 6-32 $ SID D1 D2 D3 D4 D5 D6 D7

Static Aeroelastic Analysis Sample Problems

AEFACT 1 0. .09 .21 .33 .45 .56 .66 $ D8 +AE1 .74 $ AEFACT 2 .74 .82 .90 .974 $ AEFACT 3 .974 1.00 $ AEFACT 4 0. .1875 .375 .625 .750 .875 1.00 $ $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1000 $ $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * BEAM SPLINE FIT ON THE WING * $ $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT$ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR $ ATTACHMENT AND TORSIONAL FLEXIBILITIES. CID IDENTIFIES $ THE CORD2R ENTRY THAT DEFINES THE SPLINE AXIS. DHTX AND $ DTHY ARE ROTATIONAL ATTACHMENT FLEXIBILITIES (-1.0 SPECI$ FIES NO ATTACHMENTS). $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 101 1001 1001 1035 14 0. 1. 0 $ DTHX DTHY +SP1 -1.0 -1.0 $ SPLINE2 102 2001 2001 2016 14 0. 1. 0 +SP2 -1.0 -1. $ SPLINE2 103 3001 3001 3005 14 0. 1. 0 $ $ THE SET1 ENTRY DEFINES THE SETS OF POINTS TO BE USED BY THE SURFACE SPLINE FOR INTERPOLATION. $ SID G1 G2 G3 G4 G5 G6 SET1 14 1 THRU 11 $

+AE1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +SP1

+SP2 +SP3 $ $ $ $ $ $

Listing 6-7. BAH_AERO58.DAT Input File

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Chapter 6

Static Aeroelastic Analysis Sample Problems

N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA144B $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA144B $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ $ CANTILEVERED WING WITH TEN BEAM $ $ ELEMENTS AND DUMBBELL MASSES $ $ $ $ SOLUTION STATIC AEROELASTIC SOLUTION TO $ $ AN AILERON DEFLECTION USING DOUBLET $ $ LATTICE METHOD AERODYNAMICS AT MACH $ $ NO. 0.0 $ $ $ $ OUTPUT PLOTS OF THE STICK MODEL AND AERO $ $ GRID, LISTS OF RESTRAINED AND $ $ UNRESTRAINED ANTISYMMETRIC STATIC $ $ STABILITY DERIVATIVES PLUS THE $ $ STRESSES AND DEFLECTIONS FOR A $ $ TYPICAL DESIGN CONDITION $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO CEND EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS ANTISYMMETRIC, 58 BOXES, DOUBLET-LATTICE AERO AILERON ROLL, STATIC AERO SOLUTION C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS 2 SUBTI = ANTISYMMETRIC, 58 BOXES, DOUBLET-LATTICE AERO 3 LABEL = AILERON ROLL, STATIC AERO SOLUTION 4 ECHO = BOTH 5 SPC = 13 $ ANTISYMMETRIC CONSTRAINTS 6 MPC = 1 $ CONTROL SURFACE RELATIVE MOTION 7 $OUTPUT 8 SET 2 = 7 THRU 12 9 SET 3 = 11 10 DISP = 2 11 SPCF = 3 12 AEROF = ALL 13 APRES = ALL 14 TRIM = 1 15 BEGIN BULK EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS ANTISYMMETRIC, 58 BOXES, DOUBLET-LATTICE AERO AILERON ROLL, STATIC AERO SOLUTION

PAGE

2

PAGE

I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT SECTION ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $ $ ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS, AND $ $ ITS ASSOCIATED SUPERELEMENT ID. $ $ $ $ THE BAH JET TRANSPORT WING, AS SHOWN ON P.45 OF THE BOOK $ $ "AEROELASTICITY" BY BISPLINGHOFF, ASHLEY AND HALFMAN, IS $ $ ROTATED 180 DEG AROUND THE Y AXIS. THAT ORIENTATION IS $ $ RETAINED HERE. POINTS 1 THRU 10 ARE ALONG THE ONE- AND $ $ THREE-QUARTER CHORD LINES, POINT 11 IS AT THE ROOT OF THE $ $ ELASTIC AXIS (35% CHORD), AND POINT 12 IS AT THE INBOARD $ $ TRAILING EDGE OF THE AILERON. $ $ $ INCLUDE BAH_STRUCT.DAT $ INCLUDE BAH_MASS.DAT $ INCLUDE BAH_AILERON.DAT $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $ $ THE SPC ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS $ $ AND ENFORCED DISPLACEMENTS. IT LISTS THE ID, GRID POINT $ $ NO., CONSTRAINED DOFS AND VALUE OF AN ENFORCED DISPLACE$ $ MENT. $ $ $ $ SID G C D SPC 13 11 35 $ $

6-34

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10

.

Static Aeroelastic Analysis Sample Problems

$ $ $ $ $ $ $ $ SUPORT $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ AEROS $ $ $

THE SUPORT ENTRY IDENTIFIES A GRID POINT OR A SCALAR POINT AND SPECIFIES THE DOF COMPONENTS IN WHICH THE USER DESIRES REACTIONS TO BE APPLIED TO PREVENT RIGID BODY MOTION. IT THUS INVOKES THE SOLUTION OF THE BALANCE EQUATIONS TO DETERMINE THE REACTIONS. IN THE STATIC AEROELASTIC SOLUTION THE DOF COMPONENTS MUST BE CONSISTENT WITH THE UNDEFINED VARIABLES ON THE TRIM ENTRIES. 11

4 * * * AERODYNAMIC DATA * * * (LB-IN-SEC SYSTEM) * * ELEMENT GEOMETRY * *

THE AEROS ENTRY IS UNIQUE TO THE STATIC AEROELASTICITY SOLUTION, SOL144. ACSID IDENTIFIES THE AERO COORDINATE SYSTEM. RCSID IDENTIFIES THE REFERENCE COORDINATE SYSTEM FOR RIGID BODY MOTION. REFC IS THE REFERENCE CHORD. REFB IS THE REFERENCE SPAN. REFS IS THE REFERENCE WING AREA. SYMXZ AND SYMXY ARE SYMMETRY KEYS. ACS 1

$ $ $ $ $ $ $ $

RCID 1

CHORD 162.5

SPAN 1000.0

AREA SYMYZ 81250.0 -1

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $

SYMXY

* CONTROL SURFACE DEFINITION *

EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS ANTISYMMETRIC, 58 BOXES, DOUBLET-LATTICE AERO AILERON ROLL, STATIC AERO SOLUTION

$ $ $ PAGE

I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 $ $ THE AESURF ENTRY DEFINES AN AERODYNAMIC CONTROL SURFACE. $ LISTED ARE THE ALPHANUMERIC NAME OF THE SURFACE, THE ID $ OF A COORDINATE SYSTEM THAT DEFINES THE HINGE LINE AND $ THE ID OF AN AELIST ENTRY. $ $ ID LABEL CID1 ALID1 CID2 ALID2 AESURF 503 AILE 10 2005 $ $ THE CORD2R ENTRY DEFINES THE COORDINATE SYSTEM IN WHICH THE $ HINGE-LINE IS DEFINED. IT LISTS THE ORIGIN, A POINT ALONG $ THE Z-AXIS AND A POINT IN THE X-Z PLANE. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 10 -90.0 0. 0. -90.0 0. 1. $ C1 C2 C3 +CR10 410.0 -50.0 0.0 $ INCLUDE BAH_AERO58.DAT $ $ THE AELIST ENTRY LISTS AERODYNAMIC BOXES THAT LIE ON THE $ CONTROL SURFACE. $ $ SID E1 E2 E3 ETC AELIST 2005 2005 2006 2011 2012 2017 2018 $ $ * BEAM SPLINE FIT ON THE AILERON * $ $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $ GRID POINTS ON THE SETG ENTRY TO THE SUB-REGION DEFINED $ BY AERODYNAMIC BOXES 2005 THRU 2018 OF THE REGION ON THE $ CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE $ SPLINE IS TO BE IMPOSED. $ $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 104 2001 2005 2018 15 $ $ THE SET1 ENTRY DEFINES THE SETS OF POINTS TO BE USED BY $ THE SURFACE SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 G5 G6 SET1 15 8 10 12 $ $ $ * * * SOLUTION SPECIFICATIONS * * * $ $ * * AERODYNAMIC DOFS * * $ $ THE AESTAT ENTRY LISTS TRIM VARIABLES USED TO SPECIFY $ RIGID BODY MOTIONS. THESE AND THE CONTROL SURFACE $ ROTATIONS MAKE UP THE VARIABLES IN THE EQUATIONS OF $ MOTION. $ AESTAT 501 ROLL

.. $ $ $ $ $ $

8

10

.

$ $ $ $ $ +CR10 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

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Chapter 6

Static Aeroelastic Analysis Sample Problems

AESTAT 502 URDD4 $ $ * * TRIM CONDITIONS * * $ $ THE TRIM ENTRY SPECIFIES CONSTRAINTS FOR THE TRIM VARIABLES $ LISTED ON THE AESTAT AND AESURF ENTRIES. LISTED ARE ITS ID, $ THE MACH NUMBER, DYNAMIC PRESSURE AND PAIRS OF TRIM VARI$ ABLES AND THEIR CONSTRAINED VALUES. THOSE THAT ARE NOT $ HELD FIXED MUST BE CONSTRAINED BY REACTION FORCES STIPU$ LATED ON THE SUPORT ENTRY. SEE SECTION 3.5.3 OF THE THEO$ RETICAL MANUAL FOR MORE DETAILS. $ $ TRIM CONDITION 1: STEADY ROLL $ $ ID MACH Q LABEL1 UX1 LABEL2 UX2 TRIM 1 0.0 4.0075 URDD4 0.0 AILE 1.0 $ $ THE PARAM,AUNITS,GINV PERMITS THE ACCELERATIONS ON THE TRIM $ ENTRY TO BE SPECIFIED IN UNITS OF LOAD FACTOR (I.E., IN G’S) $ PARAM AUNITS .0025907 $ ENDDATA INPUT BULK DATA CARD COUNT = 430

Listing 6-8. Input Files for Jet Transport Wing in Roll

6-36

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $

Static Aeroelastic Analysis Sample Problems

EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS PAGE 12 ANTISYMMETRIC, 58 BOXES, DOUBLET-LATTICE AERO AILERON ROLL, STATIC AERO SOLUTION S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 1 0. .09 .21 .33 .45 .56 .66 +AE1 2+AE1 .74 3AEFACT 2 .74 .82 .90 .974 4AEFACT 3 .974 1.00 5AEFACT 4 0. .1875 .375 .625 .750 .875 1.00 6AELIST 2005 2005 2006 2011 2012 2017 2018 7AEROS 1 1 162.5 1000.0 81250.0 -1 8AESTAT 501 ROLL 9AESTAT 502 URDD4 10AESURF 503 AILE 10 2005 11CAERO1 1001 1000 0 5 1 1 +CA1 12+CA1 78.75 0. 0. 225. 35. 500. 0. 100. 13CAERO1 2001 1000 0 2 4 1 +CA2 14+CA2 78.75 0. 0. 225. 35. 500. 0. 100. 15CAERO1 3001 1000 0 5 3 1 +CA3 16+CA3 78.75 0. 0. 225. 35. 500. 0. 100. 17CELAS2 3 5142661.12 5 18CMASS2 121 5248.7 1 3 19CMASS2 122 134.9 1 3 2 3 20CMASS2 123 790.3 2 3 21CMASS2 341 9727. 3 3 22CMASS2 342 11005. 3 3 4 3 23CMASS2 343 473. 4 3 24CMASS2 561 3253.6 5 3 25CMASS2 562 -139.7 5 3 6 3 26CMASS2 563 946.3 6 3 27CMASS2 781 2617.8 7 3 28CMASS2 782 21. 7 3 8 3 29CMASS2 783 782.3 8 3 30CMASS2 9101 494.8 9 3 31CMASS2 9102 -7.3 9 3 10 3 32CMASS2 9103 185.2 10 3 33CONM1 1 11 +51 34+51 17400. 4.37+7 +52 35+52 4.35+09 36CONM1 2 12 +AIL1 37+AIL1 0.0 0.0 +AIL2 38+AIL2 13970.5 39CORD2R 1 0. 0. 0. 0. 0. -1. +C1 40+C1 -1. 0. 0. 41CORD2R 10 -90.0 0. 0. -90.0 0. 1. +CR10 42+CR10 410.0 -50.0 0.0 43GENEL 432 1 3 2 3 3 3 +01 44+01 4 3 5 3 6 3 7 3 +02 45+02 8 3 9 3 10 3 +03 46+03 UD 11 3 11 4 11 5 +04 47+04 11 6 +05 48+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 49+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 50+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 51+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 52+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 53+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 54+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 55+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 56+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 57+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 58+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 59+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 60+17 413.0 61GRID 1 20.25 90. 12456 62GRID 2 -81. 90. 12456 63GRID 3 17.85 186. 12456 64GRID 4 -71.4 186. 12456 65GRID 5 15.8 268. 12456 66GRID 6 -63.2 268. 12456 67GRID 7 13.3 368. 12456 68GRID 8 -53.2 368. 12456 69GRID 9 11.05 458. 12456 70GRID 10 -44.2 458. 12456 71GRID 11 0.0 0. 126 72GRID 12 -86.45 368. 1246 73MPC 1 12 3 -1.0 8 3 1.5 +MPC1 74+MPC1 7 3 -0.5 12 5 33.25 75PAERO1 1000 76PARAM AUNITS .0025907 77PARAM GRDPNT 11 78PARAM WTMASS .0025907 79SET1 14 1 THRU 11 80SET1 15 8 10 12 81SPC 13 11 35 82SPLINE1 104 2001 2005 2018 15 83SPLINE2 101 1001 1001 1035 14 0. 1. 0 +SP1 84+SP1 -1.0 -1.0 85SPLINE2 102 2001 2001 2016 14 0. 1. 0 +SP2

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8687888990-

+SP2 SPLINE2 +SP3 SUPORT TRIM ENDDATA TOTAL COUNT=

-1.0 103 -1.0 11 1

-1.0 3001 -1.0 4 0.0

3001

3005

14

0.

1.

4.0075

URDD4

0.0

AILE

1.0

0

+SP3

91

Listing 6-9. Sorted Bulk Data Entries for Jet Transport Wing in Roll EXAMPLE HA144B: BAH JET TRANSPORT WING DYNAMIC ANALYSIS PAGE 15 ANTISYMMETRIC, 58 BOXES, DOUBLET-LATTICE AERO AILERON ROLL, STATIC AERO SOLUTION O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 11 M O * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 4.191900E+04 5.128960E+06 -1.642074E+05 0.000000E+00 * * 0.000000E+00 0.000000E+00 5.128960E+06 1.350243E+09 -2.381847E+07 0.000000E+00 * * 0.000000E+00 0.000000E+00 -1.642074E+05 -2.381847E+07 4.458796E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Y 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Z 4.191900E+04 3.917256E+00 1.223541E+02 0.000000E+00 I(S) * 7.226942E+08 3.727022E+06 0.000000E+00 * * 3.727022E+06 4.458153E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 * I(Q) * 4.458157E+09 * * 7.226906E+08 * * 0.000000E+00 * Q * 9.977400E-04 9.999995E-01 0.000000E+00 * * -9.999995E-01 9.977400E-04 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 0.0000E+00 Q = 4.0075E+00 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { 0.0000E+00 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ROLL CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX -4.414531E-01 -4.414531E-01 -5.189431E-01 -5.059597E-01 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 URDD4 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 -1.064108E-01 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 AILE CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 1.448708E-01 1.448708E-01 1.054927E-01 1.028534E-01 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 INTERMEDIATE MATRIX ... HP COLUMN 1 3.054546E-01 COLUMN 2 2.098385E+00 COLUMN 3 -7.304801E-02 A E R O S T A T I C

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NX Nastran Aeroelastic Analysis User’s Guide

O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ROLL 2.032838E-01 URDD4 0.000000E+00 AILE 1.000000E+00

Static Aeroelastic Analysis Sample Problems

A E R O S T A T I C MACH = 0.000000E+00

D A T A

R E C O V E R Y Q = 4.007500E+00

O U T P U T

T A B L E AERODYNAMIC FORCES

AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS -1.354849E-01 -5.429556E-01 1001 LS -1.071998E+03 -1.175848E+04 2 LS -4.765255E-02 -1.909676E-01 1002 LS -3.770417E+02 -4.135676E+03 3 LS -2.612218E-02 -1.046846E-01 1003 LS -2.066867E+02 -2.267095E+03 4 LS -1.495107E-02 -5.991640E-02 1004 LS -1.182974E+02 -1.297575E+03 5 LS -7.376158E-03 -2.955995E-02 1005 LS -5.836243E+01 -6.401630E+02 6 LS -3.648801E-01 -1.462257E+00 1006 LS -3.619086E+03 -3.732182E+04 7 LS -1.464361E-01 -5.868426E-01 1007 LS -1.452435E+03 -1.497824E+04 8 LS -8.362006E-02 -3.351074E-01 1008 LS -8.293911E+02 -8.553097E+03 9 LS -4.900119E-02 -1.963723E-01 1009 LS -4.860213E+02 -5.012094E+03 10 LS -2.503084E-02 -1.003111E-01 1010 LS -2.482700E+02 -2.560284E+03 11 LS -6.398618E-01 -2.564246E+00 1011 LS -5.884946E+03 -5.627479E+04 12 LS -2.556842E-01 -1.024655E+00 1012 LS -2.351582E+03 -2.248700E+04 13 LS -1.459844E-01 -5.850326E-01 1013 LS -1.342650E+03 -1.283910E+04 14 LS -8.575688E-02 -3.436707E-01 1014 LS -7.887239E+02 -7.542170E+03 15 LS -4.417319E-02 -1.770241E-01 1015 LS -4.062703E+02 -3.884959E+03 16 LS -9.208332E-01 -3.690239E+00 1016 LS -7.804856E+03 -6.878029E+04 17 LS -3.609245E-01 -1.446405E+00 1017 LS -3.059147E+03 -2.695873E+04 18 LS -2.014354E-01 -8.072526E-01 1018 LS -1.707340E+03 -1.504593E+04 19 LS -1.155547E-01 -4.630855E-01 1019 LS -9.794256E+02 -8.631187E+03 20 LS -5.842042E-02 -2.341198E-01 1020 LS -4.951635E+02 -4.363628E+03 21 LS -1.163823E+00 -4.664021E+00 1021 LS -8.304871E+03 -6.721755E+04 22 LS -4.398006E-01 -1.762501E+00 1022 LS -3.138353E+03 -2.540105E+04 23 LS -2.303485E-01 -9.231217E-01 1023 LS -1.643734E+03 -1.330397E+04 24 LS -1.207852E-01 -4.840466E-01 1024 LS -8.619053E+02 -6.976046E+03 25 LS -5.559115E-02 -2.227816E-01 1025 LS -3.966904E+02 -3.210713E+03 26 LS -1.347419E+00 -5.399781E+00 1026 LS -8.032175E+03 -5.973930E+04 27 LS -4.651038E-01 -1.863904E+00 1027 LS -2.772556E+03 -2.062088E+04 28 LS -1.955689E-01 -7.837424E-01 1028 LS -1.165817E+03 -8.670762E+03 29 LS -5.785358E-02 -2.318482E-01 1029 LS -3.448742E+02 -2.565002E+03 30 LS -1.905612E-03 -7.636740E-03 1030 LS -1.135965E+01 -8.448740E+01 31 LS -1.437025E+00 -5.758880E+00 1031 LS -6.334769E+03 -4.355154E+04 32 LS -4.041525E-01 -1.619641E+00 1032 LS -1.781605E+03 -1.224854E+04 33 LS -2.264492E-02 -9.074951E-02 1033 LS -9.982446E+01 -6.862932E+02 34 LS 2.357795E-01 9.448866E-01 1034 LS 1.039375E+03 7.145705E+03 35 LS 2.615480E-01 1.048154E+00 1035 LS 1.152969E+03 7.926661E+03 36 LS -1.530291E+00 -6.132641E+00 2001 LS -5.864338E+03 -3.504858E+04 37 LS -3.360160E-01 -1.346584E+00 2002 LS -1.287671E+03 -7.695847E+03 38 LS 2.695004E-01 1.080023E+00 2003 LS 1.377029E+03 1.097320E+04 39 LS 1.497109E+00 5.999664E+00 2004 LS 3.824786E+03 1.523938E+04 40 LS 3.661325E+00 1.467276E+01 2005 LS 9.353885E+03 3.726939E+04 41 LS 1.145043E+00 4.588762E+00 2006 LS 2.925336E+03 1.165563E+04 42 LS -1.569205E+00 -6.288589E+00 2007 LS -5.541820E+03 -3.052331E+04 43 LS -2.701145E-01 -1.082484E+00 2008 LS -9.539389E+02 -5.254117E+03 44 LS 4.182711E-01 1.676221E+00 2009 LS 1.969560E+03 1.446396E+04 45 LS 1.815431E+00 7.275338E+00 2010 LS 4.274261E+03 1.569455E+04 46 LS 4.017232E+00 1.609906E+01 2011 LS 9.458201E+03 3.472934E+04 47 LS 1.380969E+00 5.534232E+00 2012 LS 3.251362E+03 1.193859E+04 48 LS -1.562750E+00 -6.262719E+00 2013 LS -4.686912E+03 -2.370004E+04 49 LS -2.563097E-01 -1.027161E+00 2014 LS -7.687097E+02 -3.887089E+03 50 LS 3.775468E-01 1.513019E+00 2015 LS 1.509757E+03 1.017906E+04 51 LS 1.684529E+00 6.750751E+00 2016 LS 3.368097E+03 1.135417E+04 52 LS 3.921867E+00 1.571688E+01 2017 LS 7.841497E+03 2.643442E+04 53 LS 1.306940E+00 5.237563E+00 2018 LS 2.613135E+03 8.809122E+03 54 LS -1.122619E+00 -4.498896E+00 3001 LS -1.188721E+03 -6.040188E+03 55 LS -1.255417E-01 -5.031084E-01 3002 LS -1.329338E+02 -6.754702E+02 56 LS 2.223080E-01 8.908994E-01 3003 LS 2.353979E+02 1.196116E+03 57 LS 6.150346E-01 2.464751E+00 3004 LS 6.512489E+02 3.309158E+03 58 LS 8.130676E-01 3.258368E+00 3005 LS 8.609423E+02 4.374663E+03 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. POINT ID. 7 8 9 10 11 12

D I S P L A C E M E N T TYPE T1 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0

V E C T O R T2 0.0 0.0 0.0 0.0 0.0 0.0

T3 -5.400196E+00 -1.110230E+01 -1.101022E+01 -1.777396E+01 0.0 -1.563730E+01

R1 0.0 0.0 0.0 0.0 0.0 0.0

R2 0.0 0.0 0.0 0.0 0.0 -5.064504E-02

R3 0.0 0.0 0.0 0.0 0.0 0.0

Listing 6-10. Output for Jet Transport Wing in Roll

6.4 A 15-Degree Sweptback Wing in a Wind Tunnel (Example HA144C) A simple flat-plate wing with 15 deg of sweepback is shown in Figure 6-4. This wing has been tested in a wind tunnel for flutter at subsonic and supersonic speeds, and the results have been

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reported by Tuovila and McCarty (1955). A number of models of the same shape, but of different materials, were tested. The models were made of 0.041 in thick sheet metal with their leading and trailing edges beveled 0.25 in to form a symmetric hexagonal airfoil shape. The 15 deg swept model had a constant chord of 2.07055 in. and a semispan of 5.52510 in. (and an effective span with wall reflection of 11.0502 in). The low-speed models were made of aluminum, and the high-speed models were made of magnesium. The subsonic flutter test was run at Mach number m = 0.45. In the present example, the aluminum model is treated as if it were tested for its static aeroelastic characteristics in the wind tunnel at m = 0.45.

Figure 6-4. A 15-Degree Sweptback Wing Model The method of solution for the restrained wind tunnel model employs the “large mass” method discussed in Enforced Motion in the NX Nastran Basic Dynamic Analysis User’s Guide for enforced motion. A large support mass is assumed (chosen approximately six orders of magnitude larger than the system mass) which in turn is reacted by an equally large force so there is a very small consequent acceleration of the system. A gravity force is included to account for the weight of the system. The present example represents the plate model appropriately by plate elements. The basic structural model consists of GRID and CQUAD4 entries. The plate wing is divided into seven strips of equal width and four chordwise elements separated along the one-eighth, one-half, and seven-eighths chord lines; its idealization into 28 structural plate elements is shown in Figure (a). The leading- and trailing-edge CQUAD4 elements taper in thickness to zero at the edges on their corresponding CQUAD4 entries, whereas the interior elements have the constant thickness of 0.041 in. as given on the PSHELL entry. The MAT1 entry lists the moduli E = 10.3 × 106 and G = 3.9 × 106, and the weight density of ρ = 0.100 lb/in3 for aluminum. Three grid points are constrained to simulate a perfectly clamped root chord. However, the root leading and trailing edge grid points are assumed to be free because a clamp would not restrain the beveled edges. GRIDs 9 and 25 (at the one-eighth and seven-eighths chord points at the root) are connected to GRID 17 by rigid bars (RBAR 101 and RBAR 102), and GRID 17 is constrained in all degrees of freedom except the support (SUPORT) degrees of freedom, vertical translation

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(T3), and pitch (R2). GRIDs 1 through 8, 10 through 16, 18 through 24, and 26 through 40 are constrained against in-plane rotation (R3) because the CQUAD4 element has no in-plane rotational stiffness. ASET1 entries are included to reduce the size of the problem by restricting the degrees of freedom to the normal deflections of the wing (the GRID T3 degrees of freedom), which are the only deflections required to determine the aerodynamic loads. This plate model is used in several subsequent examples. The Bulk Data entries are contained in the separate input file PLATE_STRUCT. DAT that is presented in Listing 6-11. The density on the MAT1 entry and PARAM,COUPMASS,1 result in the generation of the coupled (note that a coupled mass matrix is not a consistent mass matrix) mass matrix of the plate, while the GRAV entry introduces the weight of the model into the problem. Although the weight of a wind tunnel model is usually small, it is not negligible and is just as easily included. The large mass is placed at the support point GRID 17 and is chosen arbitrarily as 105 on a CONM2 entry with its pitching moment of inertia also chosen arbitrarily as 105 lb-in2. These are converted to mass units by the parameter WTMASS. The additional PARAM,AUNITS is included to permit accelerations to be specified in units of Gs both in the input (on the TRIM entry) and in the output (the values of the UX for accelerations in the Aerostatic Data Recovery Output Table). The reacting force of 105 is also placed at GRID 17 on the FORCE entry. The LOAD entry combines the gravity load with the reacting force. (Note that the load and the force are in opposite directions.)

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Figure 6-5. A 15-Degree Sweptback Wing Model

The Aerodynamic Model The aerodynamic data begin with the AEROS entry, which includes the reference chord of 2.07055 in., the reference span (twice the exposed span) of 11.0502 in., the exposed surface area of 11.4400 sq in., and the symmetry flags SYMXZ = 1 to account for the wind tunnel wall reflection and SYMXY = 0 to neglect tunnel floor and ceiling interference. The aerodynamic

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coordinate system CORD2R has its origin at the wing support point (GRID 17) and its positive x-axis aft in the streamwise direction. The CAERO1 entry, along with a subsonic Mach number on the TRIM entry, specifies the subsonic Doublet-Lattice method. Six equal width spanwise strips are specified along with the minimum recommended number of four equal chordwise strips. The planform geometry is specified on the continuation entry. A PAERO1 entry is required although there are no interfering bodies in the problem. The division of the planform into aerodynamic boxes is shown in Figure 6-5(b).

Spline Data The aerodynamics and structure are interconnected by a surface spline SPLINE1. The 24 aerodynamic boxes are tied to the 20 grid points listed on the SET1 entries. The 20 points are variously spaced spanwise from the root to tip and along the leading edge, the one-eighth, one-half, and seven-eighth chord lines, and the trailing edge. The optional user input of downwashes caused by incidence, twist, or camber is illustrated by the direct matrix input item DMI W2GJ. The additional downwashes wj g [See Eq. 1-2] at the three-quarter chord location of each aerodynamic box are input in this manner. In this example, it is assumed that the model has no camber or twist and is set in the wind tunnel at an angle of attack of 10 deg = 0.17453 rad so that W2GJ is constant for the 24 boxes and is input using the “THRU” option on the DMI entry.

Static Aeroelastic Inputs The static aeroelastic solution requires the definition of the trim variables. Normally, both AESTAT and AESURF entries are required. However, in this example there is no trimming surface required since the model is trimmed by the inertial forces of the large mass attached to the model; therefore, no AESURF entry is input. The trim variables, angle of attack α , and pitching = ANGLEA, pitch rate qc/2V = PITCH, vertical acceleration acceleration are listed on four AESTAT entries. A TRIM entry completes the data required for the static aeroelastic solution. One entry is sufficient for both the rigid and flexible wing solutions. The entry specifies the Mach number m = 0.45, the dynamic pressure = 2.0 psi, the angle of attack α = 0.0 (since the wind tunnel setting has already been included as 10.0 deg of incidence), and the pitch rate . The ENDDATA entry completes the input Bulk Data Section.

Case Control Commands The Case Control Section begins with three title commands that appear at the top of each page of output and at the bottom of each output plot. Using ECHO = BOTH prints both the annotated and sorted Bulk Data Sections with the output. The SPC and LOAD commands invoke the corresponding Bulk Data entries (SPC1, and GRAV and FORCE, respectively). The DISP = ALL, STRESS = ALL, FORCE = ALL, AEROF = ALL, and APRES = ALL commands are print requests for structural deflections, stresses, forces, and aerodynamic forces and pressures. The TRIM command selects the single trim condition from the Bulk Data Section. The remaining commands in the Case Control Section specify output plot parameters. The OUTPUT(PLOT) command delimits the output request designated by the structure plotter. The CSCALE 2.0 command specifies spacing of characters in the titles on the plots. The PLOTTER NASTRAN command invokes the NASTRAN plotter routine. The first set of elements to be plotted is identified as the aerodynamic elements and the second set as the CQUAD4 elements. The plot titles are “AERODYNAMIC ELEMENTS” and “STRUCTURAL ELEMENTS”. The FIND command finds a scale and origin for the set to be plotted, and the PLOT command invokes

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Static Aeroelastic Analysis Sample Problems

the plot routine. For each subcase, the aerodynamic and structural elements are plotted with their elements labeled. BEGIN BULK ends the Case Control Section. In the Executive Control Section, ID NXN, HA144C is the identification of this problem. TIME 5 limits the CPU time to 5 minutes. SOL 144 calls for the static aeroelastic response analysis. CEND completes the Executive Control Section.

Output The input data file for this example is shown below in Listing 6-12 followed by the sorted Bulk Data entries in Listing 6-13 and the output results in Listing 6-14. Results of particular interest are reviewed in the following discussion. The restrained stability derivatives are found for the model and are shown in Table 6-3. The unrestrained values are not shown since they are the same as the restrained derivatives (except for the inertial derivatives which are zero in the unrestrained case), nor are the mean axis rotational derivatives shown since they are not of interest for the wind tunnel installation. Only a slight decrease in CLα is obtained along with a slight aft movement of the aerodynamic center at the high speed. The aerodynamic center in the rigid case is at chord aft of the aerodynamic reference point, which is at the midchord of the root, or at 56.95% of the root chord. In the flexible case at = 2.0 psi, the aerodynamic center has moved aft to ) and pitch acceleration ( ) derivatives provide 57.97% of the root chord. The pitch rate ( additional information about the model but have no experimental counterpart. Note that the intercept coefficients are CLo = CLα wj g and = Cmo = Cmα wj g where the incidence wj g = 0.17453 rad. Note also that the tabulated output pitch acceleration coefficients must be divided by to obtain the values listed in Table 6-3. Table 6-3. Derivatives for the 15-Degree Swept Wind Tunnel Model Rigid Value Derivative High Speed Value, = 2.0 psi 0.7829 0.6870 CLo -0.05443 -0.05477 Cmo 4.486 3.936 CLα -0.3119 -0.3138 Cmα CLq 5.204 2.930 Cmq -1.347 -1.115 0.0

0.001184

0.0

-0.0001548

0.0

-0.001662

0.0

0.0002389

Additional output data of interest include the solution trim variables, the aerodynamic pressures and forces, the grid point displacements and the structural element forces and stresses. The trim variables are the accelerations and are quite small as expected with the large support mass and the opposing applied force: = 0.00015050 and . The aerodynamic forces are substantial at the angle of attack of 10 degrees and dynamic pressure and result in large displacements that are probably

6-44

NX Nastran Aeroelastic Analysis User’s Guide

Static Aeroelastic Analysis Sample Problems

too large for the assumed small deflection theory in both the aerodynamic and structural analyses and in large stresses that are probably beyond the yield strength of the aluminum plate. The deflection of GRID 40 is 2.886 in. and the principal stresses at the root are ±78,270 psi in CQUAD4 15.

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Static Aeroelastic Analysis Sample Problems

Chapter 6

$ $ $ $ $ $ $ $ $ $ $ $ $ GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID $ $ $ $ $ $ $ $ $ CQUAD4 +M00000 CQUAD4 +M00001 CQUAD4 +M00002 CQUAD4 +M00003 CQUAD4 +M00004 CQUAD4 +M00005 CQUAD4 +M00006 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4

6-46

* * * STRUCTURAL DATA * * * (LB-IN-SEC SYSTEM) * * GRID GEOMETRY * * THE GRID ARRAY IS A FIVE BY EIGHT MESH OF EQUALLY SPACED GRIDS IN THE SPANWISE DIRECTION AND UNEQUALLY SPACED GRIDS IN THE CHORDWISE DIRECTION. CHORDWISE THE GRIDS ARE STATIONED AT THE LEADING EGDE, 1/8 CHORD, 7/8 CHORD AND TRAILING EDGE. THE WING HAS A 15 DEGREE SWEEP ANGLE. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.0 .211491 .422983 .634474 .845966 1.05746 1.26895 1.48044 .258819 .47031 .681802 .893293 1.10478 1.31628 1.52777 1.73926 1.03528 1.24677 1.45826 1.66975 1.88124 2.09273 2.30422 2.51572 1.81173 2.02322 2.23471 2.44621 2.6577 2.86919 3.08068 3.29217 2.07055 2.28204 2.49353 2.70502 2.91652 3.12801 3.3395 3.55099

0.0 .7893 1.5786 2.3679 3.1572 3.9465 4.7358 5.5251 0.0 .7893 1.5786 2.3679 3.1572 3.9465 4.7358 5.5251 0.0 .7893 1.5786 2.3679 3.1572 3.9465 4.7358 5.5251 0.0 .7893 1.5786 2.3679 3.1572 3.9465 4.7358 5.5251 0.0 .7893 1.5786 2.3679 3.1572 3.9465 4.7358 5.5251

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

* * STRUCTURAL STIFFNESS PROPERTIES * * THE ELEMENT CONNECTIVITY IS DEFINED BY THE CQUAD4 ENTRY. THE PLATES ALONG THE LEADING AND TRAILING EDGES HAVE 0.0 IN. THICKNESS BUT TAPER TO A CONTSTANT THICKNESS OF .041 IN. BETWEEN THE 1/8 AND 7/8 CHORDS. 1

1

1

2 0.0

2

1

2

3

1

3

0.0 3

0.0

0.0 4

0.0 4

1

4

5

1

5

0.0 5

0.0

0.0 6

0.0 6

1

6

7

1

7

0.0 7

0.0 0.0 8 9 10 11 12 13 14 15 16 17 18 19 20

1 1 1 1 1 1 1 1 1 1 1 1 1

0.0 8

9 10 11 12 13 14 15 17 18 19 20 21 22

0.0 10 11 12 13 14 15 16 18 19 20 21 22 23

$ $ $ $ $ $ $ $ $ $ $ $ $

10 .041 11 .041 12 .041 13 .041 14 .041 15 .041 16 .041

9

18 19 20 21 22 23 24 26 27 28 29 30 31

17 18 19 20 21 22 23 25 26 27 28 29 30

NX Nastran Aeroelastic Analysis User’s Guide

.041 10 .041 11 .041 12 .041 13 .041 14 .041 15 .041

$ $ $ $ $ $ $ $ $ +M00000 +M00001 +M00002 +M00003 +M00004 +M00005 +M00006

Static Aeroelastic Analysis Sample Problems

CQUAD4 CQUAD4 +M00007 CQUAD4 +M00008 CQUAD4 +M00009 CQUAD4 +M00010 CQUAD4 +M00011 CQUAD4 +M00012 CQUAD4 +M00013 $ $ $ $ $ $ $ $ $ PSHELL $ $ $ $ $ $ PARAM $ $ $ $ $ PARAM $

21 22

1 1

23

1

24

1

25

1

26

1

27

1

28

1

23 25 .041 26 .041 27 .041 28 .041 29 .041 30 .041 31 .041

24 26 .041 27 .041 28 .041 29 .041 30 .041 31 .041 32 .041

32 34 0.0 35 0.0 36 0.0 37 0.0 38 0.0 39 0.0 40 0.0

31 33 0.0 34 0.0 35 0.0 36 0.0 37 0.0 38 0.0 39 0.0

+M00007 +M00008 +M00009 +M00010 +M00011 +M00012 +M00013

THE PSHELL ENTRY DEFINES THE MEMBRANE, BENDING, TRANSVERSE SHEAR, AND COUPLING PROPERTIES OF THIN SHELL ELEMENTS. IT LISTS ITS ID, MATERIAL ID ENTRIES 1,2 AND 3 FOR THE MEMBRANE STIFFNESS PROPERTIES, DEFAULT VALUES OF THICKNESS 0.041, BENDING STIFFNESS AND TRANSVERSE SHEAR STIFFNESS AND THE NON-STRUCTURAL MASS DEFAULTS TO ZERO. PID 1

MID1 1

T .041

MID2 12.*I/T**3 1

MID3 1

TS/T

NSM

* * PARAMETERS * * THE PARAMETER WTMASS CONVERTS THE STRUCTURAL WEIGHT TO MASS UNITS; ITS VALUE IS 1/G.

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $

WTMASS .0025901 COUPMASS (=1), CAUSES THE GENERATION OF CONSISTENT MASS MATRICES RATHER THAN LUMPED MASS MATRICES. N V1 COUPMASS 1

$ $ $ $ $ $

Listing 6-11. PLATE_STRUCT.DAT Input File

NX Nastran Aeroelastic Analysis User’s Guide

6-47

Chapter 6

N A S T R A N

Static Aeroelastic Analysis Sample Problems

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA144C $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA144C $$$$$$$$ $ $ $ MODEL DESCRIPTION MODEL A OF NASA TN D-1824 $ $ HALF SPAN 15 DEGREE SWEPT WING $ $ 28 QUAD4 PANEL MODEL $ $ $ $ THIS RUN ALSO DEMONSTRATES THAT THE $ $ USER MAY PROVIDE DOWNWASHES DUE TO $ $ INITIAL INCDENCE, TWIST, AND CAMBER $ $ $ $ SOLUTION WIND TUNNEL MOUNT, STATIC STABILITY $ $ ANALYSIS USING DOUBLET LATTICE METHOD $ $ AERODYNAMICS AT MACH NO. 0.45 $ $ $ $ OUTPUT PLOTS OF THE STRUCTURAL MODEL $ $ LISTS OF STATIC STABILITY DERIVATIVES, $ $ AND THE STRESSES AND DEFLECTIONS OF $ $ A CANTILEVERED WING WIND TUNNEL MODEL $ $ AT AN INITIAL ANGLE OF ATTACK $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO CEND EXAMPLE HA144C: HALF SPAN 15-DEG SWEPT UNTAPERED WING CANTILEVERED WIND TUNNEL MOUNT, DOUBLET-LATTICE AERO 0.041 IN PLATE W/BEVELLED LEADING AND TRAILING EDGES C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144C: HALF SPAN 15-DEG SWEPT UNTAPERED WING 2 SUBT = CANTILEVERED WIND TUNNEL MOUNT, DOUBLET-LATTICE AERO 3 LABEL = 0.041 IN PLATE W/BEVELLED LEADING AND TRAILING EDGES 4 ECHO = BOTH 5 SPC = 1 6 LOAD = 30 $ REACTION FORCE MINUS GRAVITY LOAD 7 DISP = ALL 8 STRESS = ALL 9 FORCE = ALL 10 AEROF = ALL 11 APRES = ALL 12 TRIM = 1 13 $ 14 OUTPUT(PLOT) 15 CSCALE 2.0 16 PLOTTER NASTRAN 17 SET 1 = AERO1 18 SET 2 = QUAD4 19 PTITLE = AERODYNAMIC ELEMENTS 20 FIND SCALE, ORIGIN 1 ,SET 1 21 PLOT ORIGIN 1, SET 1, LABEL ELEMENTS 22 $ 23 PTITLE = STRUCTURAL ELEMENTS 24 FIND SCALE, ORIGIN 1, SET 2 25 PLOT ORIGIN 1, SET 2, LABEL ELEMENTS 26 BEGIN BULK EXAMPLE HA144C: HALF SPAN 15-DEG SWEPT UNTAPERED WING CANTILEVERED WIND TUNNEL MOUNT, DOUBLET-LATTICE AERO 0.041 IN PLATE W/BEVELLED LEADING AND TRAILING EDGES I N P U T B U L K

PAGE

2

PAGE D A T A

D E C K

E C H O

8 .. 9 .. . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ INCLUDE PLATE_STRUCT.DAT $ $ $ THE RBAR ENTRY DEFINES A RIGID BAR WITH SIX DEGREES OF $ $ FREEDOM AT EACH END. LISTED ARE THE GRID POINT NUMBERS, $ $ THE INDEPENDENT AND THE DEPENDENT DOFS AT THE TWO ENDS. $ $ $ $ EID GA GB CNA CNB CMA CMB $ RBAR 101 17 9 123456 RBAR 102 17 25 123456 $ $ $ * * MASS AND INERTIA PROPERTIES * * $ $ $ $ THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES FOR LINEAR, $

6-48

NX Nastran Aeroelastic Analysis User’s Guide

3

10

.

Static Aeroelastic Analysis Sample Problems

$ $ $ $ $ $ MAT1 $ $ $ $ $ $ $ $ $ $ $ CONM2 $ +CM2 $ $ $ $ $ $ $ GRAV $ $ $ $ $ $ $ FORCE $ $ $ $ $ $ LOAD $ $ PARAM

TEMPERATURE-INDEPENDENT, ISOTROPIC MATERIALS. IT LISTS YOUNGS MODULUS, THE SHEAR MODULUS, POISSONS RATIO MASS DENSITY, THERMAL EXPANSION COEFFICIENT AND STRUCTURAL ELEMENT DAMPING. MID 1

E 10.3+6

NU

RHO .100

A

TREF

GE

THE CONM2 ENTRY ASSIGNS A CONCENTRATED MASS AND MOMENT OF INERTIA TO STRUCTURAL GRID POINT G. LISTED ARE THE GRID NUMBER, THE COORDINATE SYSTEM, THE MASS AND THE LOCATION OF THE CENTER OF MASS RELATIVE TO THE GRID POINT. THE CONTINUATION ENTRY LISTS THE LOWER LEFT TRIANGLE OF THE MOMENT OF INERTIA MATRIX (SANS MINUS SIGNS). THE NUMBERS LISTED HERE ARE CHOSEN AS LARGE VALUES TO REPRESENT A GROUNDED SYSTEM. EID 53 I11

G 17 I21

CID 0 I22 1.0+5

M 1.0+5 I31

X1

X2

I32

I33

X3

THE GRAV ENTRY DEFINES A UNIT GRAVITY VECTOR TO BE USED TO APPLY THE GRAVITY LOAD. LISTED ARE THE COORDINATE SYSTEM, THE SCALAR ACCELERATION DUE TO GRAVITY AND THE COMPONENTS OF THE UNIT VECTOR. SID 10

CID

G N1 386.088 0.0

N2 0.0

$ $ $ $ $ $ $ $ $ $ $ +CM2 $ $ $ $ $ $ $

N3 -1.0 $

THE FORCE ENTRY DEFINES A STATIC LOAD AT A GRID POINT BY DEFINING A VECTOR. THE FORCE IS NECESSARY HERE TO BALANCE THE WEIGHT OF THE MODEL AND ITS SUPPORT. ID 20

G 17

CID

F 1.0+5

N1 0.0

N2 0.0

SID 30

S 1.0

S1 1.0

AUNITS

.0025901

L1 10

S2 1.0

L2 20

$ $ $ $

N3 1.0

THE LOAD ENTRY PERMITS COMBINING THE FORCE AND GRAVITY LOADS. S AND THE SI ARE SCALE FACTORS AND THE LI ARE LOAD SET ID NUMBERS. ETC

$ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ SUPORT $ $ $ $ $ $ SPC1 SPC1 SPC1 SPC1 SPC1 $ $ $ $ $ $ ASET1 ASET1 ASET1 ASET1 $ $ $ $ $ $

G 3.9+6

$ $ $ $ $

$ * * STRUCTURAL CONSTRAINTS * * THE SUPORT ENTRY IDENTIFIES A DOF IN WHICH THE USER DESIRES DETERMINATE REACTIONS TO BE APPLIED TO PREVENT RIGID BODY MOTION. IT INVOKES THE SOLUTION OF THE BALANCE EQUATIONS TO DETERMINE THE REACTIONS. IN THE STATIC AEROELASTIC SOLUTION THE DOF COMPONENTS MUST BE CONSISTENT WITH THE UNDEFINED VARIABLES ON THE TRIM ENTRIES. IT IS ALSO IMPORTANT THAT NO OTHER POINT ON THE STRUCTURE BE CONSTRAINED IN THESE DOFS; OTHERWISE, UNDETERMINED CONSTRAINT FORCES WILL BE IMPOSED AND THE REACTIONS AT THE SUPORT POINT WILL NOT BE THE TOTAL OF THE REACTION FORCES. ID 17

C 35

THE SPC1 ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS. IT LISTS ITS ID, THE DOF COMPONENTS TO BE CONSTRAINED AND THE GRID POINT NO. SID 1 1 1 1 1

C 1246 6 6 6 6

G1 17 1 10 18 26

G2

ETC.

THRU THRU THRU THRU

8 16 24 40

THE ASET1 ENTRY DEFINES DOFS THAT THE USER DESIRES TO BE INDEPENDENT. IT LISTS THE DOF COMPONENT NUMBERS AND THE GRID ID NO.S. V 3 3 3 3

ID1 1 10 18 26

THRU THRU THRU THRU THRU

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $

ID2 8 16 24 40

* * * AERODYNAMIC DATA * * * (LB-IN-SEC SYSTEM) * * ELEMENT GEOMETRY * *

$ $ $ $ $ $

NX Nastran Aeroelastic Analysis User’s Guide

6-49

Static Aeroelastic Analysis Sample Problems

Chapter 6

$ $ $ $ $ $ $ $ $ $ $ $ AEROS $ $ $ $ $ $ $ $ CORD2R $ +CORD1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

6-50

EID

PID

THE AEROS ENTRY IS UNIQUE TO THE STATIC AEROELASTICITY SOLUTION, SOL21. ACSID IDENTIFIES THE AERO COORDINATE SYSTEM. RCSID IDENTIFIES THE REFERENCE COORDINATE SYSTEM FOR RIGID BODY MOTION. REFC IS THE REFERENCE CHORD. REFB IS THE REFERENCE SPAN (NOT THE SEMI-SPAN). REFS IS THE REFERENCE WING AREA (FOR ONE SIDE ONLY, IN THIS CASE). SYMXZ AND SYMXY ARE SYMMETRY KEYS THAT SPECIFY THE PRESENCE OF REFLECTION PLANES ONLY, NOT THE EXISTENCE OF OTHER LIFTING SURFACES. ACSID 0

RCSID 11

REFC REFB REFS SYMXZ 2.07055 11.0502 11.4400 1

SYMXY

THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM FLAGGED BY THE AEROS ENTRY. THE ORIGIN IS AT THE WING SUPPORT POINT (GRID 17). LISTED ARE THE ORIGIN, A POINT 0N THE Z-AXIS AND A POINT ON THE X-AXIS, ALL IN THE RID COORDINATE SYSTEM. CID 11 C1 2.0

RID C2 0.0

A1 A2 1.035275 0.0 C3 0.0

A3 0.0

B1 B2 1.035275 0.0

B3 1.0

THE CAERO1 ENTRY IS USED FOR DOUBLET LATTICE AERODYNAMICS. LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE USED TO PARTITION THE WING INTO AERODYNAMIC PANELS. THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND EXTRA POINT IDS.

CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 101 1 0 6 4 1 $ ( FWD LEFT POINT ) ROOTCHORD ( FWD RIGHT POINT ) TIP CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1 $ $ * SURFACE SPLINE FIT ON THE WING * $ $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $ GRID POINTS ON THE SETG ENTRY TO THE SUB-REGION DEFINED $ BY AERODYNAMIC BOXES 101 THRU 124 OF THE REGION ON THE $ CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE $ SPLINE IS TO BE IMPOSED. $ $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 100 101 101 124 100 .0 $ $ THE SET1 ENTRY DEFINES THE STRUCTURAL GRID POINTS TO $ BE USED BY THE SURFACE SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 G5 G6 SET1 100 2 4 6 8 9 11 13 +S1 15 18 20 22 24 25 27 29 +S2 31 34 36 38 40 $ $ * * USER SUPPLIED AERO DATA * * $ $ THE DMI ENTRY ACCOMMODATES DIRECT INPUT OF USER SUPPLIED $ MATRICES OF DATA. LISTED ARE THE NAME OF THE MATRIX, THE $ FORM OF MATRIX, THE TYPE OF DATA BEING INPUT, AND THE TYPE $ EXPECTED AS OUTPUT. M IS THE NUMBER OF ROWS AND N IS THE $ NUMBER OF COLUMNS. THE DATA ARE EXPECTED BY COLUMNS. THE $ CONTINUATION ENTRY LISTS THE COLUMN NO., THE ROW NO. OF THE $ FIRST NON-ZERO ELEMENT AND THE FOLLOWING ELEMENTS IN THAT $ COLUMN. $ $ * INITIAL DOWNWASHES (E.G., DUE TO INCIDENCE,TWIST OR CAMBER) * $ DMI W2GJ 0 2 1 0 24 2 DMI W2GJ 1 1 0.17453 THRU 24 DMI W2GJ 2 1 0.17453 THRU 24 $ $ * * * STATIC AEROELASTIC SOLUTION * * * $ $ * * AERODYNAMIC DOFS * *

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CORD1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CA101

$ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ +S1 +S2 $ $ $ $ $ $ $ $ $ $ $ $ $ $

$ $ $ $

Static Aeroelastic Analysis Sample Problems

$ $ THE AESTAT ENTRY LISTS TRIM VARIABLES USED TO SPECIFY $ RIGID BODY MOTIONS. THESE AND THE CONTROL SURFACE $ ROTATIONS MAKE UP THE VARIABLES IN THE EQUATIONS OF $ MOTION. $ AESTAT 501 ANGLEA AESTAT 502 PITCH AESTAT 503 URDD3 AESTAT 504 URDD5 $ $ * * TRIM CONDITIONS * * $ $ THE TRIM ENTRY SPECIFIES CONSTRAINTS FOR THE TRIM VARIABLES $ LISTED ON THE AESTAT AND AESURF ENTRIES. LISTED ARE ITS ID $ THE MACH NUMBER, DYNAMIC PRESSURE AND PAIRS OF TRIM VARI$ ABLES AND THEIR CONSTRAINED VALUES. THOSE THAT ARE NOT $ HELD FIXED MUST BE CONSTRAINED BY REACTION FORCES STIPU$ LATED ON THE SUPORT ENTRY. $ TRIM 1 0.45 2.0 ANGLEA 0.0 PITCH 0.0 ENDDATA INPUT BULK DATA CARD COUNT = 352

$ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

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Chapter 6

Static Aeroelastic Analysis Sample Problems

Listing 6-12. Input Files for the 15-Degree Sweptback Wing

6-52

NX Nastran Aeroelastic Analysis User’s Guide

Static Aeroelastic Analysis Sample Problems

EXAMPLE HA144C: HALF SPAN 15-DEG SWEPT UNTAPERED WING CANTILEVERED WIND TUNNEL MOUNT, DOUBLET-LATTICE AERO 0.041 IN PLATE W/BEVELLED LEADING AND TRAILING EDGES S O R T E D B U L K D A T A E C H O CARD COUNT - . 1 .. 2 .. 3 .. 4 .. 5 . . 6 .. 7 .. 1AEROS 0 11 2.07055 11.0502 11.4400 1 2AESTAT 501 ANGLEA 3AESTAT 502 PITCH 4AESTAT 503 URDD3 5AESTAT 504 URDD5 6ASET1 3 1 THRU 8 7ASET1 3 10 THRU 16 8ASET1 3 18 THRU 24 9ASET1 3 26 THRU 40 10CAERO1 101 1 0 6 4 11+CA101 .0 .0 .0 2.07055 1.48044 5.52510 12CONM2 53 17 0 1.0+5 13+CM2 1.0+5 14CORD2R 11 1.035275 0.0 0.0 1.035275 15+CORD1 2.0 0.0 0.0 16CQUAD4 1 1 1 2 10 9 17+M00000 0.0 0.0 .041 .041 18CQUAD4 2 1 2 3 11 10 19+M00001 0.0 0.0 .041 .041 20CQUAD4 3 1 3 4 12 11 21+M00002 0.0 0.0 .041 .041 22CQUAD4 4 1 4 5 13 12 23+M00003 0.0 0.0 .041 .041 24CQUAD4 5 1 5 6 14 13 25+M00004 0.0 0.0 .041 .041 26CQUAD4 6 1 6 7 15 14 27+M00005 0.0 0.0 .041 .041 28CQUAD4 7 1 7 8 16 15 29+M00006 0.0 0.0 .041 .041 30CQUAD4 8 1 9 10 18 17 31CQUAD4 9 1 10 11 19 18 32CQUAD4 10 1 11 12 20 19 33CQUAD4 11 1 12 13 21 20 34CQUAD4 12 1 13 14 22 21 35CQUAD4 13 1 14 15 23 22 36CQUAD4 14 1 15 16 24 23 37CQUAD4 15 1 17 18 26 25 38CQUAD4 16 1 18 19 27 26 39CQUAD4 17 1 19 20 28 27 40CQUAD4 18 1 20 21 29 28 41CQUAD4 19 1 21 22 30 29 42CQUAD4 20 1 22 23 31 30 43CQUAD4 21 1 23 24 32 31 44CQUAD4 22 1 25 26 34 33 45+M00007 .041 .041 0.0 0.0 46CQUAD4 23 1 26 27 35 34 47+M00008 .041 .041 0.0 0.0 48CQUAD4 24 1 27 28 36 35 49+M00009 .041 .041 0.0 0.0 50CQUAD4 25 1 28 29 37 36 51+M00010 .041 .041 0.0 0.0 52CQUAD4 26 1 29 30 38 37 53+M00011 .041 .041 0.0 0.0 54CQUAD4 27 1 30 31 39 38 55+M00012 .041 .041 0.0 0.0 56CQUAD4 28 1 31 32 40 39 57+M00013 .041 .041 0.0 0.0 58DMI W2GJ 0 2 1 0 59DMI W2GJ 1 1 0.17453 THRU 24 60DMI W2GJ 2 1 0.17453 THRU 24 61FORCE 20 17 1.0+5 0.0 0.0 62GRAV 10 386.088 0.0 0.0 -1.0 63GRID 1 0.0 0.0 0.0 64GRID 2 .211491 .7893 0.0 65GRID 3 .422983 1.5786 0.0 66GRID 4 .634474 2.3679 0.0 67GRID 5 .845966 3.1572 0.0 68GRID 6 1.05746 3.9465 0.0 69GRID 7 1.26895 4.7358 0.0 70GRID 8 1.48044 5.5251 0.0 71GRID 9 .258819 0.0 0.0 72GRID 10 .47031 .7893 0.0 73GRID 11 .681802 1.5786 0.0 74GRID 12 .893293 2.3679 0.0 75GRID 13 1.10478 3.1572 0.0 7677787980818283-

GRID GRID GRID GRID GRID GRID GRID GRID

14 15 16 17 18 19 20 21

1.31628 1.52777 1.73926 1.03528 1.24677 1.45826 1.66975 1.88124

3.9465 4.7358 5.5251 0.0 .7893 1.5786 2.3679 3.1572

PAGE

8

..

9

..

11

10

.

0.0

1 +CA101 2.07055 +CM2

0.0

1.0

+CORD1 +M00000 +M00001 +M00002 +M00003 +M00004 +M00005 +M00006

+M00007 +M00008 +M00009 +M00010 +M00011 +M00012 +M00013 24

2

1.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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Chapter 6

Static Aeroelastic Analysis Sample Problems

84858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122-

22 2.09273 23 2.30422 24 2.51572 25 1.81173 26 2.02322 27 2.23471 28 2.44621 29 2.6577 30 2.86919 31 3.08068 32 3.29217 33 2.07055 34 2.28204 35 2.49353 36 2.70502 37 2.91652 38 3.12801 39 3.3395 40 3.55099 30 1.0 1.0 1 10.3+6 3.9+6 1 AUNITS .0025901 COUPMASS1 WTMASS .0025901 1 1 .041 101 17 9 102 17 25 100 2 4 15 18 20 31 34 36 1 6 1 1 6 10 1 6 18 1 6 26 1 1246 17 100 101 101 17 35 1 0.45 2.0 ENDDATA TOTAL COUNT= 123

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID LOAD MAT1 PAERO1 PARAM PARAM PARAM PSHELL RBAR RBAR SET1 +S1 +S2 SPC1 SPC1 SPC1 SPC1 SPC1 SPLINE1 SUPORT TRIM

3.9465 4.7358 5.5251 0.0 .7893 1.5786 2.3679 3.1572 3.9465 4.7358 5.5251 0.0 .7893 1.5786 2.3679 3.1572 3.9465 4.7358 5.5251 10

1 123456 123456 6 22 38 THRU THRU THRU THRU

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 .100

20

1 8 24 40 8 16 24 40

9 25

124

100

.0

ANGLEA

0.0

PITCH

11 27

13 29

+S1 +S2

0.0

Listing 6-13. Sorted Bulk Data Entries for the 15-Degree Sweptback Wing

6-54

NX Nastran Aeroelastic Analysis User’s Guide

Static Aeroelastic Analysis Sample Problems

EXAMPLE HA144C: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 22 CANTILEVERED WIND TUNNEL MOUNT, DOUBLET-LATTICE AERO 0.041 IN PLATE W/BEVELLED LEADING AND TRAILING EDGES N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 4.5000E-01 Q = 2.0000E+00 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ 1.0000 0.0000 0.0000 ] { X } {-1.0353E+00 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 7.829188E-01 7.829188E-01 6.870168E-01 6.870191E-01 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -5.442822E-02 -5.442822E-02 -5.476566E-02 -5.476585E-02 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ANGLEA CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 4.485869E+00 4.485869E+00 3.936382E+00 3.936395E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -3.118560E-01 -3.118560E-01 -3.137894E-01 -3.137905E-01 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 PITCH CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 5.203555E+00 5.203555E+00 2.929798E+00 2.929811E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -1.346727E+00 -1.346727E+00 -1.115142E+00 -1.115143E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 URDD3 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 1.184015E-03 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 -1.548271E-04 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+0 URDD5 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 -1.720942E-03 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 2.473614E-04 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 INTERMEDIATE MATRIX ... HP0 COLUMN 1 4.746038E-07 -5.086167E-07 INTERMEDIATE MATRIX ... HP COLUMN 1 2.722098E-06 -2.917375E-06 COLUMN 2 2.528908E-06 -2.769456E-06 COLUMN 3 -4.839799E-10 5.527028E-10 COLUMN 4 1.421871E-10 -2.102641E-10 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 0.000000E+00 PITCH 0.000000E+00 URDD3 1.504998E-04 URDD5 -2.571351E-05 A E R O S T A T I C AERODYNAMIC GRID LABEL 1 LS 2 LS 3 LS 4 LS 5 LS 6 LS 7 LS 8 LS 9 LS 10 LS 11 LS 12 LS 13 LS 14 LS 15 LS 16 LS 17 LS 18 LS 19 LS 20 LS 21 LS 22 LS 23 LS

D A T A

R E C O V E R Y O U T P U T MACH = 4.500000E-01

AERODYNAMIC PRES. AERODYNAMIC COEFFICIENTS PRESSURES 1.650690E+00 3.301379E+00 7.473637E-01 1.494727E+00 5.366659E-01 1.073332E+00 4.333381E-01 8.666762E-01 1.647448E+00 3.294896E+00 7.004328E-01 1.400866E+00 4.834796E-01 9.669591E-01 2.633696E-01 5.267392E-01 1.528239E+00 3.056479E+00 6.702324E-01 1.340465E+00 5.308363E-01 1.061673E+00 2.525186E-01 5.050372E-01 1.485955E+00 2.971910E+00 6.615313E-01 1.323063E+00 4.091381E-01 8.182762E-01 2.007488E-01 4.014975E-01 1.409623E+00 2.819246E+00 5.771903E-01 1.154381E+00 3.324486E-01 6.648972E-01 1.479836E-01 2.959672E-01 1.158140E+00 2.316279E+00 4.176423E-01 8.352845E-01 1.970198E-01 3.940396E-01

T A B L E Q = 2.000000E+00 AERODYNAMIC FORCES

EXTERNAL ID 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

LABEL LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS

NORMAL FORCES(T3) 1.573657E+00 7.124865E-01 5.116214E-01 4.131155E-01 1.570567E+00 6.677458E-01 4.609170E-01 2.510790E-01 1.456921E+00 6.389548E-01 5.060638E-01 2.407344E-01 1.416610E+00 6.306597E-01 3.900450E-01 1.913804E-01 1.343840E+00 5.502544E-01 3.169342E-01 1.410776E-01 1.104093E+00 3.981523E-01 1.878256E-01

MOMENTS(R2) 2.036459E-01 9.220243E-02 6.620859E-02 5.346103E-02 2.032461E-01 8.641257E-02 5.964698E-02 3.249197E-02 1.885393E-01 8.268677E-02 6.548940E-02 3.115328E-02 1.833227E-01 8.161329E-02 5.047549E-02 2.476642E-02 1.739056E-01 7.120809E-02 4.101427E-02 1.825676E-02 1.428801E-01 5.152465E-02 2.430640E-02

NX Nastran Aeroelastic Analysis User’s Guide

6-55

Static Aeroelastic Analysis Sample Problems

Chapter 6

24 LS 7.479374E-02 1.495875E-01 124 LS 7.130336E-02 9.227323E-03 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. D I S P L A C E M E N T V E C T O R POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 0.0 0.0 -5.254037E-03 8.232171E-02 -3.844042E-02 0.0 2 G 0.0 0.0 8.150650E-02 1.065635E-01 -3.379666E-02 0.0 3 G 0.0 0.0 3.264067E-01 4.548286E-01 -1.170699E-01 0.0 4 G 0.0 0.0 7.373135E-01 5.157071E-01 -7.268365E-02 0.0 5 G 0.0 0.0 1.219565E+00 6.523296E-01 -7.502448E-02 0.0 6 G 0.0 0.0 1.749563E+00 6.489754E-01 -4.040851E-02 0.0 7 G 0.0 0.0 2.287436E+00 6.874363E-01 -4.004871E-02 0.0 8 G 0.0 0.0 2.828604E+00 6.646019E-01 -2.491584E-02 0.0 9 G 0.0 0.0 0.0 0.0 0.0 0.0 10 G 0.0 0.0 8.946978E-02 2.012441E-01 -4.134535E-02 0.0 11 G 0.0 0.0 3.525474E-01 4.168285E-01 -8.629092E-02 0.0 12 G 0.0 0.0 7.553221E-01 5.502001E-01 -7.132266E-02 0.0 13 G 0.0 0.0 1.237087E+00 6.269942E-01 -5.915132E-02 0.0 14 G 0.0 0.0 1.759834E+00 6.642811E-01 -4.411518E-02 0.0 15 G 0.0 0.0 2.297022E+00 6.728978E-01 -3.392968E-02 0.0 16 G 0.0 0.0 2.835212E+00 6.731929E-01 -2.944477E-02 0.0 17 G 0.0 0.0 0.0 0.0 0.0 0.0 18 G 0.0 0.0 1.136017E-01 2.644884E-01 -2.274032E-02 0.0 19 G 0.0 0.0 3.981031E-01 4.300459E-01 -3.106282E-02 0.0 20 G 0.0 0.0 7.983710E-01 5.554405E-01 -3.913468E-02 0.0 21 G 0.0 0.0 1.274980E+00 6.244545E-01 -3.802212E-02 0.0 22 G 0.0 0.0 1.790500E+00 6.575523E-01 -3.456567E-02 0.0 23 G 0.0 0.0 2.322119E+00 6.695484E-01 -3.066917E-02 0.0 24 G 0.0 0.0 2.857963E+00 6.715869E-01 -2.916626E-02 0.0 25 G 0.0 0.0 0.0 0.0 0.0 0.0 26 G 0.0 0.0 1.135303E-01 2.802644E-01 2.516595E-02 0.0 27 G 0.0 0.0 4.077593E-01 4.626091E-01 7.991176E-03 0.0 28 G 0.0 0.0 8.194947E-01 5.700024E-01 -1.377494E-02 0.0 29 G 0.0 0.0 1.299275E+00 6.290278E-01 -2.346403E-02 0.0 30 G 0.0 0.0 1.814852E+00 6.592954E-01 -2.749739E-02 0.0 31 G 0.0 0.0 2.345347E+00 6.675459E-01 -2.872346E-02 0.0 32 G 0.0 0.0 2.879997E+00 6.716351E-01 -2.730966E-02 0.0 33 G 0.0 0.0 -7.731961E-03 5.269706E-02 4.167109E-02 0.0 34 G 0.0 0.0 1.057840E-01 2.622258E-01 4.648738E-02 0.0 35 G 0.0 0.0 4.047464E-01 4.993872E-01 1.622160E-02 0.0 36 G 0.0 0.0 8.222791E-01 5.558869E-01 -3.542251E-03 0.0 37 G 0.0 0.0 1.305281E+00 6.521177E-01 -2.346955E-02 0.0 38 G 0.0 0.0 1.821408E+00 6.406795E-01 -2.170623E-02 0.0 39 G 0.0 0.0 2.353268E+00 6.892233E-01 -3.290629E-02 0.0 40 G 0.0 0.0 2.886319E+00 6.465759E-01 -2.223470E-02 0.0 F O R C E S ELEMENT ID FX 1 0.0 2 0.0 3 0.0 4 0.0 5 0.0 6 0.0 7 0.0 8 0.0 9 0.0 10 0.0 11 0.0 12 0.0 13 0.0 14 0.0 15 0.0 16 0.0 17 0.0 18 0.0 19 0.0 20 0.0 21 0.0 22 0.0 23 0.0 24 0.0 25 0.0 26 0.0 27 0.0 28 0.0

I N

S T R E S ELEMENT FIBRE ID. DISTANCE 1 -1.025000E-02 1.025000E-02 2 -1.025000E-02 1.025000E-02 3 -1.025000E-02 1.025000E-02 4 -1.025000E-02 1.025000E-02 5 -1.025000E-02

6-56

Q U A D R I L A T E R A L - MEMBRANE FORCES FY FXY MX 0.0 0.0 -1.973221E+00 0.0 0.0 -5.454906E+00 0.0 0.0 -1.541650E+00 0.0 0.0 -1.924214E+00 0.0 0.0 -1.696724E-01 0.0 0.0 -4.288711E-01 0.0 0.0 2.551605E-01 0.0 0.0 -1.874362E+01 0.0 0.0 -1.439413E+01 0.0 0.0 -9.362266E+00 0.0 0.0 -5.242195E+00 0.0 0.0 -2.496411E+00 0.0 0.0 -7.013355E-01 0.0 0.0 -6.295047E-02 0.0 0.0 -2.170790E+01 0.0 0.0 -1.294998E+01 0.0 0.0 -8.524811E+00 0.0 0.0 -4.649412E+00 0.0 0.0 -2.285321E+00 0.0 0.0 -7.082699E-01 0.0 0.0 -2.058553E-01 0.0 0.0 -4.123490E+00 0.0 0.0 -3.932424E+00 0.0 0.0 -1.454395E+00 0.0 0.0 -1.475571E+00 0.0 0.0 -1.200144E-01 0.0 0.0 -5.645998E-01 0.0 0.0 4.050496E-01 S E S

E L E M E N T S ( Q U A D 4 ) - BENDING MOMENTS - TRANSVERSE SHEAR FORCES MY MXY QX QY 3.024332E-01 1.907779E-01 -3.173141E-01 1.017937E+00 -7.039570E-01 -4.921928E-02 -4.055569E-01 1.319205E+00 1.752648E-01 6.190604E-01 -2.997236E-01 7.771828E-01 -1.904920E-01 3.199919E-01 -2.112260E-01 5.165404E-01 1.003854E-01 3.506400E-01 -1.641087E-01 4.476568E-01 -1.101465E-01 1.073558E-01 -1.128355E-01 4.042867E-01 5.545839E-02 9.394334E-02 -5.126301E-02 2.210779E-01 -4.948936E+00 -1.809898E-02 -9.445847E+00 2.027260E+00 -1.338344E+00 -3.829413E-01 -7.855835E+00 1.798051E+00 2.691290E-01 8.351150E-01 -6.275851E+00 7.810127E-01 2.304201E-01 7.146806E-01 -4.608058E+00 3.886636E-01 1.953354E-01 6.688161E-01 -3.063073E+00 2.016163E-01 1.313061E-01 4.249342E-01 -1.634140E+00 1.205711E-01 5.821897E-02 1.654798E-01 -3.662618E-01 4.158391E-02 -5.413020E+00 1.909820E+00 -9.207546E+00 6.020051E-01 -6.357294E-01 2.905566E-01 -6.789074E+00 -3.532801E-01 5.491099E-03 -1.994554E-01 -5.508847E+00 -2.937697E-01 1.132636E-01 9.948941E-02 -3.980758E+00 -2.818420E-01 9.797072E-02 1.467821E-01 -2.666178E+00 -1.946682E-01 9.310451E-02 1.203176E-01 -1.374869E+00 -1.649903E-01 5.172014E-02 9.343265E-02 -3.105215E-01 -1.934092E-01 4.094767E-01 6.647069E-01 -1.501274E-01 5.086169E-01 -3.979302E-01 4.002048E-01 -5.527662E-02 -4.299713E-01 2.263509E-01 -6.734979E-02 -6.507438E-02 -1.015426E-01 -1.339186E-01 8.611768E-02 -7.926976E-02 -1.427904E-02 1.540736E-01 -1.153676E-02 -4.717677E-02 4.193060E-02 -1.163481E-01 2.379308E-02 -3.093108E-02 1.097691E-02 1.259939E-01 1.435611E-02 -1.664330E-02 7.441904E-02

I N Q U A D R I L A T E R A L E L E M E N T S ( Q U A D 4 ) STRESSES IN ELEMENT COORD SYSTEM PRINCIPAL STRESSES (ZERO SHEAR) NORMAL-X NORMAL-Y SHEAR-XY ANGLE MAJOR MINOR VON MISES -1.408605E+04 2.158952E+03 1.361889E+03 85.2409 2.272333E+03 -1.419943E+04 1.546135E+04 1.408605E+04 -2.158952E+03 -1.361889E+03 -4.7591 1.419943E+04 -2.272333E+03 1.546135E+04 -3.894043E+04 -5.025272E+03 -3.513571E+02-89.4065-5.021633E+03 -3.894407E+04 3.669189E+04 3.894043E+04 5.025272E+03 3.513571E+02 0.5935 3.894407E+04 5.021633E+03 3.669189E+04 -1.100524E+04 1.251146E+03 4.419229E+03 72.1017 2.678372E+03 -1.243246E+04 1.396562E+04 1.100524E+04 -1.251146E+03 -4.419229E+03-17.8983 1.243246E+04 -2.678372E+03 1.396562E+04 -1.373621E+04 -1.359847E+03 2.284297E+03 79.8695-9.516965E+02 -1.414436E+04 1.369334E+04 1.373621E+04 1.359847E+03 -2.284297E+03-10.1305 1.414436E+04 9.516965E+02 1.369334E+04 -1.211225E+03 7.166118E+02 2.503082E+03 55.5307 2.434960E+03 -2.929573E+03 4.652400E+03

NX Nastran Aeroelastic Analysis User’s Guide

Static Aeroelastic Analysis Sample Problems

1.025000E-02 6 -1.025000E-02 1.025000E-02 7 -1.025000E-02 1.025000E-02 8 -2.050000E-02 2.050000E-02 9 -2.050000E-02 2.050000E-02 10 -2.050000E-02 2.050000E-02 11 -2.050000E-02 2.050000E-02 12 -2.050000E-02 2.050000E-02 13 -2.050000E-02 2.050000E-02 14 -2.050000E-02 2.050000E-02 15 -2.050000E-02 2.050000E-02 16 -2.050000E-02 2.050000E-02 17 -2.050000E-02 2.050000E-02 18 -2.050000E-02 2.050000E-02 19 -2.050000E-02 2.050000E-02 20 -2.050000E-02 2.050000E-02

1.211225E+03 -3.061542E+03 3.061542E+03 1.821490E+03 -1.821490E+03 -6.690167E+04 6.690167E+04 -5.137702E+04 5.137702E+04 -3.341677E+04 3.341677E+04 -1.871099E+04 1.871099E+04 -8.910451E+03 8.910451E+03 -2.503280E+03 2.503280E+03 -2.246894E+02 2.246894E+02 -7.748209E+04 7.748209E+04 -4.622241E+04 4.622241E+04 -3.042764E+04 3.042764E+04 -1.659516E+04 1.659516E+04 -8.157003E+03 8.157003E+03 -2.528031E+03 2.528031E+03

-7.166118E+02 -7.862926E+02 7.862926E+02 3.958957E+02 -3.958957E+02 -1.766426E+04 1.766426E+04 -4.776955E+03 4.776955E+03 9.606033E+02 -9.606033E+02 8.224392E+02 -8.224392E+02 6.972113E+02 -6.972113E+02 4.686713E+02 -4.686713E+02 2.078012E+02 -2.078012E+02 -1.932071E+04 1.932071E+04 -2.269111E+03 2.269111E+03 1.959940E+01 -1.959940E+01 4.042723E+02 -4.042723E+02 3.496873E+02 -3.496873E+02 3.323183E+02 -3.323183E+02

-2.503082E+03-34.4693 2.929573E+03 7.663711E+02 73.0167-5.522342E+02 -7.663711E+02-16.9833 3.295601E+03 6.706247E+02 21.6270 2.087375E+03 -6.706247E+02-68.3730-1.300116E+02 -6.460077E+01-89.9248-1.766417E+04 6.460077E+01 0.0752 6.690176E+04 -1.366834E+03-88.3214-4.736899E+03 1.366834E+03 1.6786 5.141707E+04 2.980779E+03 85.0809 1.217145E+03 -2.980779E+03 -4.9191 3.367331E+04 2.550912E+03 82.6811 1.150073E+03 -2.550912E+03 -7.3189 1.903862E+04 2.387208E+03 76.7877 1.257665E+03 -2.387208E+03-13.2123 9.470904E+03 1.516719E+03 67.2067 1.106033E+03 -1.516719E+03-22.7933 3.140641E+03 5.906477E+02 55.0542 6.205446E+02 -5.906477E+02-34.9458 6.374327E+02 6.816729E+03 83.4038-1.853245E+04 -6.816729E+03 -6.5962 7.827036E+04 1.037085E+03 88.6491-2.244655E+03 -1.037085E+03 -1.3509 4.624687E+04 -7.119167E+02-88.6613 3.623633E+01 7.119167E+02 1.3387 3.044428E+04 3.551079E+02 88.8038 4.116870E+02 -3.551079E+02 -1.1962 1.660258E+04 5.239100E+02 86.4890 3.818324E+02 -5.239100E+02 -3.5110 8.189149E+03 4.294501E+02 81.6431 3.954041E+02 -4.294501E+02 -8.3569 2.591116E+03

S T R E S S E S I N Q U A D R I L A T E R A L E L E M E N T S ELEMENT FIBRE STRESSES IN ELEMENT COORD SYSTEM ID. DISTANCE NORMAL-X NORMAL-Y SHEAR-XY ANGLE 21 -2.050000E-02 -7.347600E+02 1.846049E+02 3.334895E+02 72.0200 2.050000E-02 7.347600E+02 -1.846049E+02-3.334895E+02 -17.9800 22 -1.025000E-02 -2.943598E+04 2.923094E+03 4.745082E+03 81.8274 1.025000E-02 2.943598E+04 -2.923094E+03-4.745082E+03 -8.1726 23 -1.025000E-02 -2.807203E+04 -2.840667E+03 2.856905E+03 83.6201 1.025000E-02 2.807203E+04 2.840667E+03-2.856905E+03 -6.3799 24 -1.025000E-02 -1.038235E+04 1.615831E+03-4.807837E+02 -87.7090 1.025000E-02 1.038235E+04 -1.615831E+03 4.807837E+02 2.2910 25 -1.025000E-02 -1.053353E+04 -9.559919E+02 6.147603E+02 86.3423 1.025000E-02 1.053353E+04 9.559919E+02-6.147603E+02 -3.6577 26 -1.025000E-02 -8.567358E+02 1.099871E+03-8.235638E+01 -87.5940 1.025000E-02 8.567358E+02 -1.099871E+03 8.235638E+01 2.4060 27 -1.025000E-02 -4.030456E+03 -8.305634E+02 1.698495E+02 86.9701 1.025000E-02 4.030456E+03 8.305634E+02-1.698495E+02 -3.0299 28 -1.025000E-02 2.891490E+03 8.994210E+02 1.024826E+02 2.9373 1.025000E-02 -2.891490E+03 -8.994210E+02-1.024826E+02 -87.0627

-2.434960E+03 -3.295601E+03 5.522342E+02 1.300116E+02 -2.087375E+03 -6.690176E+04 1.766417E+04 -5.141707E+04 4.736899E+03 -3.367331E+04 -1.217145E+03 -1.903862E+04 -1.150073E+03 -9.470904E+03 -1.257665E+03 -3.140641E+03 1.106033E+03 -6.374327E+02 -6.205446E+02 -7.827036E+04 1.853245E+04 -4.624687E+04 2.244655E+03 -3.044428E+04 -3.623633E+01 -1.660258E+04 -4.116870E+02 -8.189149E+03 -3.818324E+02 -2.591116E+03 -3.954041E+02

4.652400E+03 3.057124E+03 3.057124E+03 2.025501E+03 2.025501E+03 6.005085E+04 6.005085E+04 4.921988E+04 4.921988E+04 3.429809E+04 3.429809E+04 1.963893E+04 1.963893E+04 1.015830E+04 1.015830E+04 3.815834E+03 3.815834E+03 1.089473E+03 1.089473E+03 7.084602E+04 7.084602E+04 4.516639E+04 4.516639E+04 3.046241E+04 3.046241E+04 1.681220E+04 1.681220E+04 8.386587E+03 8.386587E+03 2.809763E+03 2.809763E+03

( Q U A D 4 ) PRINCIPAL STRESSES (ZERO SHEAR) MAJOR MINOR VON MISES 2.928337E+02 -8.429888E+02 1.021390E+03 8.429888E+02 -2.928337E+02 1.021390E+03 3.604554E+03 -3.011744E+04 3.207200E+04 3.011744E+04 -3.604554E+03 3.207200E+04 -2.521229E+03 -2.839147E+04 2.721857E+04 2.839147E+04 2.521229E+03 2.721857E+04 1.635065E+03 -1.040159E+04 1.130813E+04 1.040159E+04 -1.635065E+03 1.130813E+04 -9.166931E+02 -1.057282E+04 1.014558E+04 1.057282E+04 9.166931E+02 1.014558E+04 1.103331E+03 -8.601962E+02 1.704805E+03 8.601962E+02 -1.103331E+03 1.704805E+03 -8.215731E+02 -4.039446E+03 3.697757E+03 4.039446E+03 8.215731E+02 3.697757E+03 2.896749E+03 8.941626E+02 2.569146E+03 -8.941626E+02 -2.896749E+03 2.569146E+03

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Listing 6-14. Output for the 15-Degree Sweptback Wing

6.5

FSW Airplane in Antisymmetric Maneuvers (Example HA144D)

The FSW airplane of Example HA144A is reconsidered here for its lateral-directional stability characteristics and loading (Figure 6-6). The half-span model is modified to add a sweptback fin and to consider antisymmetrical motions plus the effects of a 25% chord aileron and 25% chord rudder. The right-half fin is assumed to weigh 50 Ibs. and is idealized as shown in the side view in Figure 6-6(b) with a sweepback angle of 30 deg and no taper. Two weights are located at the one-quarter and three-quarter chord of the fin at its midspan location and are assumed to be connected to its 50% chord elastic axis by rigid streamwise bars. The weights per side are assumed to be 30 lbs forward and 20 Ibs aft, yielding a fin centroid at 45% of its chord. Since the fin is assumed to have the same stiffness properties as the wing, then the right-side fin has half of the wing section properties. The aileron is shown on the wing in the plan view of Figure 6-6(a).

Figure 6-6. Idealization of Half-Span FSW Configuration Lateral-Directional Model The Bulk Data Section is only slightly different from that of Example HA144A with the addition of the fin aileron, and rudder as the new control surfaces. The SPC1 constraints are modified to permit the antisymmetric degrees of freedom. GRIDs 310, 311, and 312 are added for the fin,

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CBAR 310 provides the fin elastic axis, and RBARs 311 and 312 connect the fin masses CONM2 311 and 312 to the elastic axis. The aerodynamic reference geometry and coordinates are given on the AEROS entry. This entry lists: •

CORD2R This is the NACA coordinate system and is used as a reference for the stability derivatives.

•

the reference chord of 10.0 ft

•

the reference span of 40.0 ft

•

the reference area of the half-model of 200.0 sq. ft

•

SYMXZ = -1 for antisymmetric motion

The aerodynamic model also requires the addition of the fin using CAERO1 3100 and its spline SPLlNE2 3100 with its axis on CORD2R 300 and connected to the SET1 3100 grid points. The control surfaces are defined by AESURF and AELIST entries along with the coordinate systems whose y-axes define their hinge lines. AESURF 517 is the aileron and includes the trailing edge boxes on the outboard half of the wing as enumerated on AELIST 2000. The aileron rotation is assumed to be positive trailing-edge up on the right wing as given by CORD2R 110; this definition of positive aileron gives a positive rolling moment (right wing down). AESURF 518 is the rudder and includes the four trailing edge boxes on the fin as listed on AELIST 3000. A trailing-edge left rudder rotation gives a positive side force from the rudder so the rudder hinge line is defined by CORD2R 301. For use in a later example, the above data are separated into the input file HA144D_MODEL.DAT as shown in Listing 6-15, and included in the present example using the INCLUDE entry. The AESTAT entry defines the motions that respond to the control surface inputs. In the lateral-directional case these are: sideslip, β = SIDES; roll rate, pb/2V = ROLL; yaw rate, rb/2V ; and the rotational accelerations in roll, = YAW; the lateral (side) acceleration, , and yaw, . These six motions are defined on AESTAT entries 511 through 516. PARAM,AUNITS permits load factor units for the accelerations. Two subsonic trim conditions are considered. The first, TRIM 1, uses the sea-level value of dynamic pressure = 1200 psf at a Mach number of m = 0.9 and finds the steady roll solution for an aileron rotation of δa = AILERON = 25 deg = 0.436332 rad, no yaw rate, rb/2V = YAW = 0.0, no lateral (side) acceleration, , and no rotational accelerations, and . The second trim condition, TRlM 2, is an abrupt roll at high speed, and assumes zero roll and yaw rates, pb/2V = YAW = 0.0 and rb/2V = YAW 0.0, and no side or yaw accelerations, URDD2 = URDD6 = 0.0. The ENDDATA entry completes the Bulk Data Section. The Case Control Section begins with three title commands. Using ECHO = BOTH prints the input data with and without the annotations. SPC = 1 enforces the constraints in the Bulk Data Section. Using DISP = STRESS = FORCE = AEROF = APRES = ALL calculates and prints all of these derived quantities from the analysis. The two subcases request that the two input trim cases be analyzed. The OUTPUT(PLOT) and remaining entries request plots of the deformed structure overlaid on the undeformed structure. The Case Control Section ends with BEGIN BULK.

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The Executive Control Section begins with the identification ID NXN, HA144D. TIME 5 limits the total computing time to 5.0 CPU minutes. SOL 144 calls for the Static Aeroelastic Response DMAP sequence. The Executive Control Section is concluded with the CEND entry. The input data file is shown in Listing 6-16; the sorted Bulk Data are in Listing 6-17, and those are followed by selected output in Listing 6-18. Highlights of the selected output are discussed next. The OUTPUT FROM THE GRID POINT WEIGHT GENERATOR shows the effect of the additional fin weight. The half-airplane now weighs 8,050 lbs, and the centroid is now 2.276 ft aft of GRID 90 and 2.484 ft outboard from the centerline. The restrained lateral-directional stability derivatives are defined by:

Side Force

Equation 6-5.

Rolling Moment

Equation 6-6.

Yawing Moment

Equation 6-7. The rotations of the mean axes in the lateral-directional case are defined in a similar manner to those in the longitudinal case defined by Rodden and Love (1985). However, in the lateral-directional case there are two sets of rotations that are required in the equations of motion when using restrained aeroelastic derivatives: one is the rotation of the mean x-axis in the xy-plane, and the other is the rotation of the mean y-axis in the yz-plane. The rotation of the mean x-axis in the xy-plane is

Equation 6-8.

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The rotation of the mean y-axis in the yz-plane is

Equation 6-9. The derivatives are printed in the table NONDlMENSlONAL STABILITY AND CONTROL DERIVATIVE COEFFIClENTS and the mean axis rotations are printed in the second (for the angle βm) and third (for the angle γm) rows of INTERMEDIATE MATRIX... HP and are listed in Table 6-4 for the two restraint conditions. Note that the angular acceleration derivatives and rotations are divided by b/2, because of their definitions in Eq. 6-6 through Eq. 6-9, just as the pitch acceleration terms are divided by in Example HA144A . Note also that the values of βmβ and γmβ are obtained by adding 1.0 to the tabulated rotations. The trim solutions follow the printouts of the stability derivatives. The steady roll trim solution for δa = 25 deg = 0.436332 rad is

The trim requirement for positive rudder is consistent with the lifting surface theory predicting proverse aileron yaw. To predict adverse yaw from the aileron requires the additional ability to predict induced drag on the lifting surfaces. It should also be noted that the predicted dihedral effect Clβ does not include the effects from the sweep of the planform. The second trim solution is the abrupt roll at high speed. The solution is

The pressure data and aerodynamic box forces and moments are shown for the high-speed maneuvers. These forces, moments, and data are followed by deflections relative to the support point at GRID 90. Finally, the forces and stresses in the CBAR elements are presented. Table 6-4. Lateral-Directional Derivatives of FSW Airplane UnrestrainedValue at RestrainedValue at Rigid Value Derivative = 1200 psf = 1200 psf C yβ -0.7158 -0.6697 -0.7260 Clβ -0.3276 -0.02825 -0.02707 C nβ 0.2592 0.2425 0.2630 -0.1082 -0.1142 -0.1026 CYδa 0.2748 0.2993 0.2625 Clδa 0.03948 0.04149 0.03753 C nδ a

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Table 6-4. Lateral-Directional Derivatives of FSW Airplane UnrestrainedValue at RestrainedValue at Rigid Value Derivative = 1200 psf = 1200 psf 0.3491 0.2997 0.3381 CYδa 0.03745 0.03508 0.03229 Clδr -0.1707 -0.1526 -0.1665 Cnδr CYp 0.07965 0.1210 0.09466 Clp -0.4185 -0.5070 -0.4448 Cnp -0.02605 -0.04054 -0.03138 0.7233 0.6676 0.7285 CYr 0.04299 0.03858 0.03630 Clr -0.2775 -0.2573 -0.2794 Cnr

βmβ βmδa βmδr βmp βmr

γmβ γmδa γmδr γmp γmr

6-62

-

0.0007188

-

-

0.00003018

-

-

-0.0002602

-

-

0.0001011

-

-

-0.0001818

-

-

-0.00003559

-

-

-0.001107

-

-

-0.00004991

-

-

0.0004040

-

-

1.006878 -0.2378 -0.01859 0.4273 -0.01403

-

-

0.00001049

-

-

0.001214

-

-

-0.000001260

-

-

-0.9630 -0.006602 0.02767 0.006442 0.04077

-

1.0

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Table 6-4. Lateral-Directional Derivatives of FSW Airplane UnrestrainedValue at RestrainedValue at Rigid Value Derivative = 1200 psf = 1200 psf -

-0.0003089

-

-

0.00000291

-

-

0.0008229

-

$*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ * * * STRUCTURAL DATA * * * $ $ $ $ (LB-FT-SEC SYSTEM) $ $ $ $ * * GRID GEOMETRY * * $ $ $ $ GRID 90 - 100 (T3) FUSELAGE POINTS $ $ GRID 110 - 122 (T3) WING POINTS $ $ GRID 310 - 312 (t3) FIN POINTS $ $ $ $ * FUSELAGE GRID * $ $ $ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $ $ ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS AND $ $ ITS ASSOCIATED SUPERELEMENT ID. $ $ $ $ ID CP X1 X2 X3 CD PS SEID GRID 90 15. 0. 0. GRID 97 0. 0. 0. GRID 98 10. 0. 0. GRID 99 20. 0. 0. GRID 100 30. 0. 0. $ $ $ * WING GRID * $ $ $ $ ID CP X1 X2 X3 CD PS SEID GRID 111 24.61325 +5. 0. GRID 110 27.11325 +5. 0. GRID 112 29.61325 +5. 0. GRID 121 18.83975+15. 0. GRID 120 21.33975+15. 0. GRID 122 23.83975+15. 0. $ $ $ * FIN GRID * $ $ $ GRID 310 32.88675+0. 5. GRID 311 30.38675+0. 5. GRID 312 35.38675+0. 5. $ $ $ * * STRUCTURAL STIFFNESS PROPERTIES * * $ $ $ $ * FUSELAGE STRUCTURE * $ $ $ $ THE CBAR ENTRY DEFINES A SIMPLE BEAM ELEMENT. LISTED ARE $ $ ITS PROPERTY ENTRY ID, THE TWO GRID POINTS JOINED BY THE $ $ BEAM AND COMPONENTS OF A VECTOR FROM THE FIRST POINT. $ $ THIS VECTOR DEFINES THE DIRECTION OF THE STRUCTURAL DE$ $ FLECTION OF THE POINT AND ITS POSITIVE SENSE. $ $ $ $ EID PID GA GB X1,GO X2 X3 CBAR 101 100 97 98 0. 0. 1. CBAR 102 100 98 90 0. 0. 1. CBAR 100 100 90 99 0. 0. 1. CBAR 103 100 99 100 0. 0. 1. $ $ $ THE PBAR ENTRY DEFINES GEOMETRIC PROPERTIES OF THE BEAM. $ $ LISTED ARE ITS ASSOCIATED MATERIAL ENTRY ID, ITS CROSS SEC$ $ TIONAL AREA, AREA MOMENTS OF INERTIA, TORSIONAL MOMENT $ $ OF INERTIA AND NON-STRUCTURAL MASS PER UNIT AREA. THE $ $ OPTIONAL CONTINUATION ENTRY CONTAINS STRESS RECOVERY $ $ COEFFICIENTS, I.E., Y,Z COORDINATES WHERE STRESSES ARE $ $ TO BE COMPUTED. K1 AND K2 ARE AREA FACTORS FOR SHEAR $

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Chapter 6

$ $ $ $ $ PBAR $ +PB1F $ +PB2F $ $ $ $ CBAR CBAR $ $ $ $ $ $ $ $ RBAR RBAR RBAR RBAR $ PBAR +PB1W +PB2W $ $ $ CBAR $ RBAR RBAR $ PBAR +PB1FI +PB2FI $ $ $ $ $ $ $ MAT1 $ $ $ $ $ $ $ $ $ $ $ CONM2 CONM2 CONM2 CONM2 $ $ $ CONM2 CONM2 CONM2 CONM2 $ $ $ CONM2 CONM2 $ $ $ $ $ $ $ $ $ PARAM $ $ $

6-64

STIFFNESS (DEFAULT IS BLANK; THEN SHEAR STIFFNESS IS INFINITE, I.E., SHEAR FLEXIBILITY IS ZERO. I12 IS THE AREA PRODUCT OF INERTIA. PID 100 C1 1.0 K1

MID 1 C2 1.0 K2

A 2.0 D1 1.0 I12 0.0

I1 .173611 D2 -1.0

I2 0.15 E1 -1.0

J 0.5 E2 1.0

$ $ $ $

NSM +PB1F F1 -1.0

F2 -1.0

$ $ $

* WING STRUCTURE * EID 110 120

PID 101 101

GA 100 110

GB 110 120

X1,GO 0. 0.

X2 0. 0.

X3 1. 1.

THE RBAR ENTRY DEFINES A RIGID BAR. LISTED ARE THE GRID POINTS AT EACH END AND THE DEPENDENT AND INDEPENDENT DOFS AT EACH END. THE NUMBER OF INDEPENDENT DOFS AT THE TWO ENDS MUST EQUAL SIX. BY DEFAULT THOSE NOT DECLARED INDEPENDENT ARE MADE DEPENDENT. EID 111 112 121 122

GA 110 110 120 120

GB 111 112 121 122

CNA 123456 123456 123456 123456

CNB

101 0.5

1 3.0

1.5 0.5 0.0

0.173611+2.0 -3.0 -0.5

CMA

301

100

310

0.

311 312

310 310

311 312

123456 123456

301 0.5

1 3.0

.75 0.5 0.

.086806 1.0 -3.0 -0.5

$ $ $ $ $ $ $

CMB

0.462963 3.0 -0.5

-3.0

$ +PB1W +PB2W $ $ $

* FIN STRUCTURE * 310

+PB2F

0.

1. $

.231482 3.0 -0.5

-3.0

THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES. LISTED ARE ITS ID, ITS ELASTIC MODULUS, SHEAR MODULUS, POISSONS RATIO, MASS DENSITY, TEMPERATURE EXPANSION COEFFICIENT, REFERENCE TEMPERATURE AND A STRUCTURAL DAMPING COEFFICIENT. MID 1

E 1.44+9

G 5.40+8

NU

RHO

A

TREF

* FUSELAGE MASSES * THE CONM2 ENTRY DEFINES A CONCENTRATED MASS. LISTED ARE ITS ID, GRID LOCATION, COORDINATE SYSTEM TO LOCATE THE CENTER OF GRAVITY, THE MASS VALUE AND THE LOCATION OF THE CENTER OF GRAVITY RELATIVE TO THE GRID LOCATION. G 97 98 99 100

CID 0 0 0 0

M 1500.0 1500.0 1500.0 1500.0

X1

X2

111 112 121 122

0 0 0 0

311 312

0 0

$ $ $

30.0 20.0

* * STRUCTURAL PARAMETERS * * THE PARAM,GRDPNT,XX ENTRY CAUSES THE GRID POINT WEIGHT GENERATOR TO BE EXECUTED USING GRID POINT XX AS THE REFERENCE POINT. THEN THE INERTIA MATRIX, THE TRANSFER MATRIX FROM BASIC TO PRINCIPAL AXES AND OTHER PERTINENT INERTIA DATA ARE PRINTED. GRDPNT

$ $ $

600.0 400.0 600.0 400.0 * FIN MASSES *

311 312

$ $ $ $ $ $ $ $ $ $

X3

* WING MASSES * 111 112 121 122

$ $ $ $ $

GE

* * MASS AND INERTIA PROPERTIES * *

EID 97 98 99 100

$ +PB1FI +PB2FI

$ $ $ $ $ $ $ $ $

90

THE PARAM,WTMASS,GINV CAUSES ALL THE STRUCTURAL MASSES AND MASS DENSITIES TO BE MULTIPLIED BY GINV (I.E., BY ONE OVER

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$ $ $

Static Aeroelastic Analysis Sample Problems

$ THE ACCELERATION OF GRAVITY). THE DYNAMIC PRESSURE SUPPLIED $ FOR AERODYNAMIC FORCE CALCULATIONS WILL NOT BE MULTIPLIED $ BY GINV. PARAM WTMASS .031081 $ * * STRUCTURAL CONSTRAINTS * * $ $ THE SPC1 ENTRY CONSTRAINS THE LISTED GRID POINTS IN THE $ SPECIFIED DOF COMPONENTS. $ SID C G1 G2 G3 G4 SPC1 1 135 90 SPC1 1 35 97 98 99 100 $ $ THE SUPORT ENTRY IDENTIFIES A GRID POINT OR A SCALAR POINT $ AND SPECIFIES THE DOF COMPONENTS IN WHICH THE USER DESIRES $ REACTIONS TO BE APPLIED TO PREVENT RIGID BODY MOTION. IT $ THUS INVOKES THE SOLUTION OF THE BALANCE EQUATIONS TO DETER$ MINE THE REACTIONS. IN THE STATIC AEROELASTIC SOLUTION $ THE DOF COMPONENTS MUST BE CONSISTENT WITH THE UNDEFINED $ VARIABLES ON THE TRIM ENTRIES. $ ID C SUPORT 90 246 $ $ THE OMIT1 ENTRY IDENTIFIES GRID POINTS TO BE OMITTED FROM $ THE REMAINDER OF THE ANALYSIS. $ $ ID G G OMIT1 4 110 120 310 $ $ THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM $ FLAGGED BY THE AEROS ENTRY. THE ORIGIN IS AT THE CANARD $ QUARTER CHORD. LISTED ARE THE ORIGIN, A POINT ALONG THE $ Z AXIS AND A POINT IN THE X-Z PLANE, ALL IN THE RID $ COORDINATE SYSTEM. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0 12.5 0. 0. 12.5 0. 10. $ C1 C2 C3 +CRD1 20. 0. 0. $ THIS CORD2R ENTRY DEFINES THE NACA COORDINATE SYSTEM TO $ WHICH ALL THE STABILITY DERIVATIVES AND TRIM CONDITIONS $ WILL BE REFERENCED. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 -10.0 $ C1 C2 C3 +CRD100 0.0 0.0 0.0 $ * WING AERODYNAMIC MODEL * $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AERODYNAMICS. $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS, $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ EXTRA POINT IDS. $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 1100 1000 8 4 1 $ ( FWD LEFT POINT ) CHORD ( FWD RIGHT POINT ) CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CAW 25. 0. 0. 10. 13.45299+20. 0. 10. $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1000 $ * CANARD AERODYNAMIC MODEL * $ CAERO1 1000 1000 2 4 1 +CAC 10. 0. 0. 10. 10. 5. 0. 10. $ * FIN AERODYNAMIC MODEL * $ CAERO1 3100 1000 4 4 1 +CA1FI 30.7735 0. 10. 10. 25. 0. 0. 10. $ $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * BEAM SPLINE FIT ON THE WING * $ $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT$ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR

$ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CRD1 $ $ $ $ +CRD100 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CAW

$ $ $ $ $ $ +CAC $ $ +CA1FI $ $ $ $ $ $ $ $ $ $

NX Nastran Aeroelastic Analysis User’s Guide

6-65

Static Aeroelastic Analysis Sample Problems

Chapter 6

$ $ $ $ $ $ SPLINE2 $ +SPW $ $ $ $ $ SET1 $ $ $ SPLINE2 +SPC $ SET1 $ $ $ SPLINE2 +SP2FI $ SET1 $ $ $ $ $ $ $ $ $ CORD2R $ +CRD2 $ $ $ CORD2R +CRD2FI $ $ $ $ $ $ $ $ AESURF AESURF $ $ $ AELIST AELIST $ $ $ $ CORD2R +CRD2A $ $ CORD2R +CRD2R

6-66

ATTACHMENT AND TORSIONAL FLEXIBILITIES. CID IDENTIFIES THE CORD2R ENTRY THAT DEFINES THE SPLINE AXIS. DTHX AND DTHY ARE ROTATIONAL ATTACHMENT FLEXIBILITIES (-1. SPECIFIES NO ATTACHMENT). EID 1601 DTHX -1.

CAERO 1100 DTHY -1.

ID1 1100

ID2 1131

SETG 1100

DZ 0.

DTOR 1.

CID 2

THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. SID 1100

G1 99

G2 100

G3 111

G4 112

121

1000 -1.

1000

1000

98

99

1007

1000

0.

+SPW $ $ $ $

122

* BEAM SPLINE FIT ON THE CANARD * 1501 1.

$ $ $ $ $

1.

1

$ $ $ +SPC $

* BEAM SPLINE FIT ON THE FIN * 3100 -1.

3100 -1.

3100

3115

3100

3100

99

100

311

312

0.

1.

300

$ $ $ +SP2FI $

THE CORD2R ENTRY DEFINES THE COORDINATE SYSTEM IN WHICH THE BEAM SPLINE EXTENDS ALONG THE WING Y-AXIS. IT LISTS THE ORIGIN, A POINT ALONG THE Z-AXIS AND A POINT IN THE X-Z PLANE. * WING SPLINE AXIS * CID CS 2 0 C1 C2 38.66025+5.0

A1 30. C3 0.

A2 0.

A3 0.

B1 30.

B2 0.

B3 10.

+CRD2

0.

$ $ $ +CRD2FI

* FIN SPLINE AXIS * 300 20.0

0 0.

30.0 5.7735

0.

* CONTROL SURFACE THE AESURF ENTRY DEFINES AN LISTED ARE THE ALPHANUMERIC OF A COORDINATE SYSTEM THAT THE ID OF AN AELIST ENTRY. ID LABEL CID1 517 AILERON 110 518 RUDDER 301 THE AELIST ENTRY LISTS CONTROL SURFACE. SID E1 E2 2000 1119 1123 3000 3103 3107

0.

30.0

10.0

DEFINITION * AERODYNAMIC CONTROL SURFACE. NAME OF THE SURFACE, THE ID DEFINES THE HINGE LINE AND

ALID1 CID2 ALID2 2000 3000 AERODYNAMIC BOXES THAT LIE ON THE E3 1127 3111

* AILERON * 110 0 26.7265 10.0 0. 36.7265 15.7735 0. 0 0.

32.5 5.7735

$ $ $ $ $ $ $

$ $

ETC 1131 3115

26.7265 10.0

-10.0

$ $ $ $ +CRD2A

32.5

0.0

$ $ +CRD2R

* CONTROL SURFACE HINGE LINES *

301 22.5

$ $ $ $ $ $ $ $

* RUDDER * 0. 0.

NX Nastran Aeroelastic Analysis User’s Guide

-10.

Static Aeroelastic Analysis Sample Problems

Listing 6-15. HA144D_MODEL.DAT Input File N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA144D $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA144D $$$$$$$$ $ $ $ MODEL DESCRIPTION 30 DEG FORWARD SWEPT WING WITH $ $ AILERON, CANARD AND AFT SWEPT $ $ VERTICAL FIN AND RUDDER. $ $ BAR MODEL WITH DUMBBELL MASSES. $ $ $ $ SOLUTION ANTISYMMETRIC STATIC STABILITY $ $ DERIVATIVE ANALYSIS USING DOUBLET$ $ LATTICE AERODYNAMICS AT MACH 0.9. $ $ $ $ OUTPUT PLOTS OF THE STICK MODEL AND AERO $ $ GRID, LISTS OF RESTRAINED AND $ $ UNRESTRAINED SYMMETRIC STATIC $ $ STABILITY DERIVATIVES, THE $ $ AERODYNAMIC FORCES AND PRESSURES, $ $ AND STRESSES AND DEFLECTIONS FOR $ $ ROLL MANEUVERS. $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO CEND EXAMPLE HA144D: 30 DEG FWD SWEPT WING WITH CANARD & FIN PAGE 2 ANTISYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144D: 30 DEG FWD SWEPT WING WITH CANARD & FIN 2 SUBTI = ANTISYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO 3 LABEL = HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING 4 ECHO = BOTH 5 SPC = 1 $ SYMMETRIC CONSTRAINTS 6 DISP = ALL $ PRINT ALL DISPLACEMENTS 7 STRESS = ALL $ PRINT ALL STRESSES 8 FORCE = ALL $ PRINT ALL FORCES 9 AEROF = ALL $ PRINT ALL AERODYNAMIC FORCES 10 APRES = ALL $ PRINT ALL AERODYNAMIC PRESSURES 11 SUBCASE 1 12 TRIM = 1 $ HIGH SUBSONIC SPEED STEADY ROLL 13 SUBCASE 2 14 TRIM = 2 $ HIGH SUBSONIC SPEED ABRUPT ROLL 15 OUTPUT(PLOT) 16 PLOTTER = NASTRAN 17 SET 1 = ALL 18 FIND SCALE, ORIGIN 1, SET 1 19 PLOT SET 1 20 PLOT STATIC DEFORMATION 0, ORIGIN 1, SET 1, OUTLINE 21 BEGIN BULK EXAMPLE HA144D: 30 DEG FWD SWEPT WING WITH CANARD & FIN PAGE ANTISYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING I N P U T B U L K D A T A D E C K E C H O

3

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 INCLUDE HA144D_MODEL.DAT $ $ $ THE PARAM,AUNITS,GINV PERMITS THE ACCELERATIONS ON THE TRIM $ $ ENTRY TO BE SPECIFIED IN UNITS OF LOAD FACTOR, I.E., IN G’S. $ $ $ PARAM AUNITS .031081 $ * * AERODYNAMIC DOFS * * $ $ $ $ THE AESTAT ENTRY LISTS TRIM VARIABLES USED TO SPECIFY $ $ RIGID BODY MOTIONS. THESE AND THE CONTROL SURFACE $ $ ROTATIONS MAKE UP THE VARIABLES IN THE EQUATIONS OF $ $ MOTION. $ $ $ $ ID LABEL AESTAT 511 SIDES AESTAT 512 YAW AESTAT 513 ROLL AESTAT 514 URDD2 AESTAT 515 URDD4 AESTAT 516 URDD6 $ $ $ * * * AERODYNAMIC DATA * * * $ $ $ $ (LB-FT-SEC SYSTEM) $ $ $ $ * * ELEMENT GEOMETRY * * $ $ $ $ THE AEROS ENTRY IS UNIQUE TO THE STATIC AEROELASTICITY $ $ SOLUTION, SOL21. ACSID IDENTIFIES THE AERO COORDINATE $ $ SYSTEM, RCSID IDENTIFIES THE REFERENCE COORDINATE SYS$ $ TEM FOR RIGID BODY MOTION. REFC IS THE REFERENCE CHORD. $ $ REFB IS THE REFERENCE SPAN. REFS IS THE REFERENCE WING $ 6-67 NX Nastran Aeroelastic Analysis User’s Guide $ AREA. SYMXZ AND SYMXY ARE SYMMETRY KEYS. $ $ $

Chapter 6

Static Aeroelastic Analysis Sample Problems

$ $ $ $ $ $ $ $ $ $ $ $ TRIM +TR1 $

* * TRIM CONDITIONS * *

$ $ THE TRIM ENTRY SPECIFIES CONSTRAINTS FOR THE TRIM VARIABLES $ LISTED ON THE AESTAT AND AESURF ENTRIES. LISTED ARE ITS ID, $ THE MACH NUMBER, DYNAMIC PRESSURE AND PAIRS OF TRIM VARI$ ABLES AND THEIR CONSTRAINED VALUES. THOSE THAT ARE NOT $ HELD FIXED MUST BE CONSTRAINED BY REACTION FORCES STIPU$ LATED ON THE SUPORT ENTRY. SEE SECTION 3.5.3 OF THE THEO$ RETICAL MANUAL FOR MORE DETAILS. $ $ TRIM CONDITION 1: STEADY ROLL AT HIGH DYNAMIC PRESSURE $ $ 1 0.9 1200.0 AILERON .436332 YAW 0. +TR1 URDD2 0. URDD4 0. URDD6 0. $ TRIM CONDITION 2: ABRUPT ROLL AT HIGH DYNAMIC PRESSURE $ $ 2 0.9 1200.0 AILERON .436332 YAW 0. +TR2 URDD2 0. ROLL 0. URDD6 0. * * * $

$ $ TRIM +TR2 $ ENDDATA INPUT BULK DATA CARD COUNT =

399

Listing 6-16. Input Files for FSW Airplane in Antisymmetric Maneuvers EXAMPLE HA144D: 30 DEG FWD SWEPT WING WITH CANARD & FIN ANTISYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 1AELIST 2000 1119 1123 1127 1131 2AELIST 3000 3103 3107 3111 3115 3AEROS 1 100 10.0 40.0 200.0 -1 4AESTAT 511 SIDES 5AESTAT 512 YAW 6AESTAT 513 ROLL 7AESTAT 514 URDD2 8AESTAT 515 URDD4 9AESTAT 516 URDD6 10AESURF 517 AILERON 110 2000 11AESURF 518 RUDDER 301 3000 12CAERO1 1000 1000 2 4 13+CAC 10. 0. 0. 10. 10. 5. 0. 14CAERO1 1100 1000 8 4 15+CAW 25. 0. 0. 10. 13.45299+20. 0. 16CAERO1 3100 1000 4 4 17+CA1FI 30.7735 0. 10. 10. 25. 0. 0. 18CBAR 100 100 90 99 0. 0. 1. 19CBAR 101 100 97 98 0. 0. 1. 20CBAR 102 100 98 90 0. 0. 1. 21CBAR 103 100 99 100 0. 0. 1. 22CBAR 110 101 100 110 0. 0. 1. 23CBAR 120 101 110 120 0. 0. 1. 24CBAR 310 301 100 310 0. 0. 1. 25CONM2 97 97 0 1500.0 26CONM2 98 98 0 1500.0 27CONM2 99 99 0 1500.0 28CONM2 100 100 0 1500.0 29CONM2 111 111 0 600.0 30CONM2 112 112 0 400.0 31CONM2 121 121 0 600.0 32CONM2 122 122 0 400.0 33CONM2 311 311 0 30.0 34CONM2 312 312 0 20.0 35CORD2R 1 0 12.5 0. 0. 12.5 0. 36+CRD1 20. 0. 0. 37CORD2R 2 0 30. 0. 0. 30. 0. 38+CRD2 38.66025+5.0 0. 39CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 40+CRD100 0.0 0.0 0.0 41CORD2R 110 0 26.7265 10.0 0. 26.7265 10.0 42+CRD2A 36.7265 15.7735 0. 43CORD2R 300 0 30.0 0. 0. 30.0 10.0 44+CRD2FI 20.0 0. 5.7735 45CORD2R 301 0 32.5 0. 0. 32.5 -10. 46+CRD2R 22.5 0. 5.7735 47GRID 90 15. 0. 0. 48GRID 97 0. 0. 0. 49GRID 98 10. 0. 0. 50GRID 99 20. 0. 0. 51GRID 100 30. 0. 0. 52GRID 110 27.11325+5. 0. 53GRID 111 24.61325+5. 0. 54GRID 112 29.61325+5. 0. 55GRID 120 21.33975+15. 0. 56GRID 121 18.83975+15. 0. 57GRID 122 23.83975+15. 0. 58GRID 310 32.88675+0. 5. 59GRID 311 30.38675+0. 5. 60GRID 312 35.38675+0. 5. 61MAT1 1 1.44+9 5.40+8 62OMIT1 4 110 120 310 63- Aeroelastic PAERO1 1000 NX Nastran AnalysisGRDPNT User’s Guide 6-68 64PARAM 90 65PARAM WTMASS .031081

PAGE

..

9

1 10. 1 10. 1 10.

11

..

10

+CAC +CAW +CA1FI

10.

+CRD1

10.

+CRD2

-10.0

+CRD100

-10.0

+CRD2A

0.

+CRD2FI

0.0

+CRD2R

.

Static Aeroelastic Analysis Sample Problems

85868788899091929394959697-

SPC1 1 SPC1 1 SPLINE2 1501 +SPC 1. SPLINE2 1601 +SPW -1. SPLINE2 3100 +SP2FI -1. SUPORT 90 TRIM 1 +TR1 URDD2 TRIM 2 +TR2 URDD2 ENDDATA TOTAL COUNT=

35 135 1000 -1. 1100 -1. 3100 -1. 246 0.9 0. 0.9 0.

97 90 1000

98

99

100

1007

1000

0.

1.

1

+SPC

1100

1131

1100

0.

1.

2

+SPW

3100

3115

3100

0.

1.

300

+SP2FI

1200.0 URDD4 1200.0 ROLL

AILERON 0. AILERON 0.

.436332 URDD6 .436332 URDD6

YAW 0. YAW 0.

0.

+TR1

0.

+TR2

98

Listing 6-17. Sorted Bulk Data Entries for FSW Airplane in Antisymmetric Maneuvers EXAMPLE HA144D: 30 DEG FWD SWEPT WING WITH CANARD & FIN PAGE 15 ANTISYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 90 M O * 8.050000E+03 0.000000E+00 0.000000E+00 0.000000E+00 2.500000E+02 -2.000000E+04 * * 0.000000E+00 8.050000E+03 0.000000E+00 -2.500000E+02 0.000000E+00 1.832234E+04 * * 0.000000E+00 0.000000E+00 8.050000E+03 2.000000E+04 -1.832234E+04 0.000000E+00 * * 0.000000E+00 -2.500000E+02 2.000000E+04 2.512500E+05 -1.456625E+05 -4.346688E+03 * * 2.500000E+02 0.000000E+00 -1.832234E+04 -1.456625E+05 9.476353E+05 0.000000E+00 * * -2.000000E+04 1.832234E+04 0.000000E+00 -4.346688E+03 0.000000E+00 1.196385E+06 * S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 8.050000E+03 0.000000E+00 2.484472E+00 3.105590E-02 Y 8.050000E+03 2.276067E+00 0.000000E+00 3.105590E-02 Z 8.050000E+03 2.276067E+00 2.484472E+00 0.000000E+00 I(S) * 2.015528E+05 1.001412E+05 3.777671E+03 * * 1.001412E+05 9.059246E+05 -6.211180E+02 * * 3.777671E+03 -6.211180E+02 1.104993E+06 * I(Q) * 9.198783E+05 * * 1.105014E+06 * * 1.875778E+05 * Q * 1.380458E-01 4.794522E-03 9.904142E-01 * * -9.904069E-01 -5.532546E-03 1.380715E-01 * * 6.141498E-03 -9.999732E-01 3.984783E-03 * HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING SUBCASE 1 N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 9.0000E-01 Q = 1.2000E+03 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { 1.5000E+01 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 SIDES CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -7.158447E-01 -7.158447E-01 -6.697496E-01 -7.259881E-01 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX -3.276105E-02 -3.276105E-02 -2.824561E-02 -2.706753E-02 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 2.592296E-01 2.592296E-01 2.425404E-01 2.630050E-01 YAW CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 7.233017E-01 7.233017E-01 6.675829E-01 7.284513E-01 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 4.298554E-02 4.298554E-02 3.858262E-02 3.630278E-02 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ -2.775080E-01 -2.775080E-01 -2.572732E-01 -2.794162E-01 ROLL CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 7.965025E-02 7.965025E-02 1.209624E-01 9.465972E-02 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX -4.184672E-01 -4.184672E-01 -5.070223E-01 -4.448231E-01 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ -2.605253E-02 -2.605253E-02 -4.053827E-02 -3.137640E-02 URDD2 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 7.188179E-04 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 3.018458E-05 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 -2.601771E-04 0.000000E+00 URDD4 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 2.021708E-03 0.000000E+00 6-69 NX Nastran Aeroelastic Analysis User’s Guide

Chapter 6

Static Aeroelastic Analysis Sample Problems

CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ

URDD6

AILERON

RUDDER

A E R O S T A T I C

D A T A

0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -3.636955E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -7.117986E-04 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.213319E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -9.981033E-04 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 8.080994E-03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.082398E-01 -1.082398E-01 -1.142208E-01 -1.026430E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 2.747558E-01 2.747558E-01 2.993375E-01 2.624515E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.948340E-02 3.948340E-02 4.149461E-02 3.752508E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.491402E-01 3.491402E-01 2.996780E-01 3.381154E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.745069E-02 3.745069E-02 3.507515E-02 3.228768E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.706690E-01 -1.706690E-01 -1.525649E-01 -1.665393E-01 INTERMEDIATE MATRIX ... HP COLUMN 1 -1.678953E-01 6.878061E-03 -3.704068E-02 COLUMN 2 1.784256E-01 -1.403152E-02 4.077220E-02 COLUMN 3 3.155028E-02 4.272658E-01 6.442384E-03 COLUMN 4 -5.583723E-03 1.048801E-05 -3.089367E-04 COLUMN 5 3.735903E-04 2.427399E-02 5.812574E-05 COLUMN 6 1.905984E-02 -2.519723E-05 1.645743E-02 COLUMN 7 -3.053800E-02 -2.377972E-01 -6.602198E-03 COLUMN 8 1.061561E-01 -1.858679E-02 2.767085E-02 R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX SIDES -1.823546E-02 YAW 0.000000E+00 ROLL 2.600432E-01 URDD2 0.000000E+00 URDD4 0.000000E+00 URDD6 0.000000E+00 AILERON 4.363320E-01 RUDDER 2.058725E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC PRES. AERODYNAMIC PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 8.344750E+01 1000 LS 5.215469E+02 3.259668E+02 7.135028E+00 1001 LS 4.459393E+01 2.787120E+01 6.954690E+00 1002 LS 4.346682E+01 2.716676E+01 1.409345E+01 1003 LS 8.808403E+01 5.505252E+01 1.967786E+02 1004 LS 1.229866E+03 7.686666E+02 1.482418E+01 1005 LS 9.265111E+01 5.790694E+01 1.684706E+01 1006 LS 1.052941E+02 6.580883E+01 3.740932E+01 1007 LS 2.338083E+02 1.461302E+02 1.564302E+02 1100 LS 9.776888E+02 6.110555E+02 3.312442E+01 1101 LS 2.070276E+02 1.293923E+02 -3.004207E+01 1102 LS -1.877630E+02 -1.173519E+02 -6.053308E+00 1103 LS -3.783317E+01 -2.364573E+01 3.884376E+02 1104 LS 2.427735E+03 1.517334E+03 -3.465986E+01 1105 LS -2.166241E+02 -1.353901E+02 -7.916637E+01 1106 LS -4.947899E+02 -3.092437E+02 -4.833408E+01 1107 LS -3.020880E+02 -1.888050E+02 1.008616E+03 1108 LS 6.303850E+03 3.939906E+03 -4.184470E+00 1109 LS -2.615294E+01 -1.634559E+01 -1.988246E+02 1110 LS -1.242654E+03 -7.766587E+02 -1.356671E+02 1111 LS -8.479193E+02 -5.299496E+02 1.286082E+03 1112 LS 8.038014E+03 5.023759E+03 8.501208E+01 1113 LS 5.313256E+02 3.320785E+02 -3.836247E+02 1114 LS -2.397655E+03 -1.498534E+03 -3.771199E+02 1115 LS -2.357000E+03 -1.473125E+03 1.518862E+03 1116 LS 9.492890E+03 5.933056E+03 1.928606E+02 1117 LS 1.205379E+03 7.533619E+02 -4.639574E+02 1118 LS -2.899734E+03 -1.812333E+03 -1.745346E+03 1119 LS -1.090842E+04 -6.817759E+03 1.655755E+03 1120 LS 1.034847E+04 6.467791E+03 3.147179E+02 1121 LS 1.966987E+03 1.229367E+03 -4.135921E+02 1122 LS -2.584950E+03 -1.615594E+03 -1.965191E+03 1123 LS -1.228244E+04 -7.676525E+03 1.622306E+03 1124 LS 1.013941E+04 6.337134E+03 4.060495E+02 1125 LS 2.537810E+03 1.586131E+03 -2.535266E+02 1126 LS -1.584541E+03 -9.903381E+02 -1.928696E+03 1127 LS -1.205435E+04 -7.533970E+03

GRID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

AERODYNAMIC LABEL COEFFICIENTS LS 6.953958E-02 LS 5.945857E-03 LS 5.795575E-03 LS 1.174454E-02 LS 1.639822E-01 LS 1.235348E-02 LS 1.403922E-02 LS 3.117443E-02 LS 1.303585E-01 LS 2.760369E-02 LS -2.503506E-02 LS -5.044424E-03 LS 3.236980E-01 LS -2.888322E-02 LS -6.597198E-02 LS -4.027840E-02 LS 8.405132E-01 LS -3.487058E-03 LS -1.656872E-01 LS -1.130559E-01 LS 1.071735E+00 LS 7.084341E-02 LS -3.196873E-01 LS -3.142666E-01 LS 1.265719E+00 LS 1.607172E-01 LS -3.866311E-01 LS -1.454455E+00 LS 1.379795E+00 LS 2.622649E-01 LS -3.446601E-01 LS -1.637659E+00 LS 1.351922E+00 LS 3.383746E-01 LS -2.112721E-01 LS -1.607247E+00

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37 LS 1.081452E+00 1.297742E+03 1128 LS 8.110890E+03 5.069306E+03 38 LS 3.072515E-01 3.687018E+02 1129 LS 2.304386E+03 1.440241E+03 39 LS -1.031863E-02 -1.238236E+01 1130 LS -7.738976E+01 -4.836860E+01 40 LS -1.353353E+00 -1.624024E+03 1131 LS -1.015015E+04 -6.343843E+03 41 LS -1.499667E-01 -1.799601E+02 3100 LS -1.124750E+03 -7.029689E+02 42 LS -1.346411E-02 -1.615693E+01 3101 LS -1.009808E+02 -6.311300E+01 43 LS 4.820608E-03 5.784729E+00 3102 LS 3.615456E+01 2.259660E+01 44 LS 3.093417E-02 3.712100E+01 3103 LS 2.320063E+02 1.450040E+02 45 LS -7.612907E-02 -9.135488E+01 3104 LS -5.709680E+02 -3.568550E+02 46 LS -3.047888E-02 -3.657465E+01 3105 LS -2.285916E+02 -1.428698E+02 47 LS 6.683536E-04 8.020244E-01 3106 LS 5.012652E+00 3.132908E+00 48 LS 3.554533E-02 4.265440E+01 3107 LS 2.665900E+02 1.666188E+02 49 LS 4.053394E-02 4.864073E+01 3108 LS 3.040046E+02 1.900029E+02 50 LS -2.538108E-02 -3.045730E+01 3109 LS -1.903581E+02 -1.189738E+02 51 LS -6.749813E-03 -8.099775E+00 3110 LS -5.062360E+01 -3.163975E+01 52 LS 3.387398E-02 4.064878E+01 3111 LS 2.540549E+02 1.587843E+02 53 LS 1.469073E-01 1.762888E+02 3112 LS 1.101805E+03 6.886282E+02 54 LS -4.172111E-03 -5.006533E+00 3113 LS -3.129083E+01 -1.955677E+01 55 LS -1.380020E-02 -1.656024E+01 3114 LS -1.035015E+02 -6.468845E+01 56 LS 2.685823E-02 3.222987E+01 3115 LS 2.014367E+02 1.258979E+02 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING SUBCASE 2 N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 9.0000E-01 Q = 1.2000E+03 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { 1.5000E+01 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 SIDES CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -7.158447E-01 -7.158447E-01 -6.697496E-01 -7.259881E-01 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX -3.276105E-02 -3.276105E-02 -2.824561E-02 -2.706753E-02 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 2.592296E-01 2.592296E-01 2.425404E-01 2.630050E-01 YAW CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 7.233017E-01 7.233017E-01 6.675829E-01 7.284513E-01 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 4.298554E-02 4.298554E-02 3.858262E-02 3.630278E-02 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ -2.775080E-01 -2.775080E-01 -2.572732E-01 -2.794162E-01 ROLL CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 7.965025E-02 7.965025E-02 1.209624E-01 9.465972E-02 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX -4.184672E-01 -4.184672E-01 -5.070223E-01 -4.448231E-01 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ -2.605253E-02 -2.605253E-02 -4.053827E-02 -3.137640E-02 URDD2 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 7.188179E-04 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 3.018458E-05 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 -2.601771E-04 0.000000E+00 URDD4 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 2.021708E-03 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 -3.636955E-03 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 -7.117986E-04 0.000000E+00 URDD6 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 -2.213319E-02 0.000000E+00 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 -9.981033E-04 0.000000E+00 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 8.080994E-03 0.000000E+00 AILERON CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -1.082398E-01 -1.082398E-01 -1.142208E-01 -1.026430E-01 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 2.747558E-01 2.747558E-01 2.993375E-01 2.624515E-01 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ 3.948340E-02 3.948340E-02 4.149461E-02 3.752508E-02 RUDDER CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 3.491402E-01 3.491402E-01 2.996780E-01 3.381154E-01 CZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMX 3.745069E-02 3.745069E-02 3.507515E-02 3.228768E-02 CMY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMZ -1.706690E-01 -1.706690E-01 -1.525649E-01 -1.665393E-01 INTERMEDIATE MATRIX ... HP COLUMN 1 -1.678953E-01 6.878061E-03 -3.704068E-02 COLUMN 2

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1.784256E-01

-1.403152E-02 4.077220E-02 COLUMN 3 4.272658E-01 6.442384E-03 COLUMN 4 -5.583723E-03 1.048801E-05 -3.089367E-04 COLUMN 5 3.735903E-04 2.427399E-02 5.812574E-05 COLUMN 6 1.905984E-02 -2.519723E-05 1.645743E-02 COLUMN 7 -3.053800E-02 -2.377972E-01 -6.602198E-03 COLUMN 8 1.061561E-01 -1.858679E-02 2.767085E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX SIDES -6.297442E-02 YAW 0.000000E+00 ROLL -6.938894E-18 URDD2 0.000000E+00 URDD4 4.454218E+00 URDD6 0.000000E+00 AILERON 4.363320E-01 RUDDER 1.099768E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS -4.527578E-04 -5.433094E-01 1000 LS -3.395684E+00 -2.122302E+00 2 LS -5.883921E-04 -7.060705E-01 1001 LS -4.412941E+00 -2.758088E+00 3 LS -1.556158E-03 -1.867390E+00 1002 LS -1.167119E+01 -7.294493E+00 4 LS -4.046254E-03 -4.855505E+00 1003 LS -3.034690E+01 -1.896681E+01 5 LS -1.045061E-03 -1.254073E+00 1004 LS -7.837959E+00 -4.898724E+00 6 LS -1.345259E-03 -1.614311E+00 1005 LS -1.008944E+01 -6.305902E+00 7 LS -3.623927E-03 -4.348712E+00 1006 LS -2.717945E+01 -1.698716E+01 8 LS -1.008748E-02 -1.210498E+01 1007 LS -7.565610E+01 -4.728506E+01 9 LS -6.170879E-02 -7.405054E+01 1100 LS -4.628159E+02 -2.892599E+02 10 LS -3.456980E-02 -4.148376E+01 1101 LS -2.592735E+02 -1.620459E+02 11 LS -2.932927E-02 -3.519513E+01 1102 LS -2.199696E+02 -1.374810E+02 12 LS -5.151585E-03 -6.181902E+00 1103 LS -3.863688E+01 -2.414805E+01 13 LS -1.462438E-01 -1.754925E+02 1104 LS -1.096828E+03 -6.855176E+02 14 LS -1.315322E-01 -1.578387E+02 1105 LS -9.864918E+02 -6.165573E+02 15 LS -9.292479E-02 -1.115097E+02 1106 LS -6.969359E+02 -4.355850E+02 16 LS -3.714078E-02 -4.456893E+01 1107 LS -2.785558E+02 -1.740974E+02 17 LS -1.884638E-01 -2.261566E+02 1108 LS -1.413479E+03 -8.834241E+02 18 LS -2.229333E-01 -2.675199E+02 1109 LS -1.672000E+03 -1.045000E+03 19 LS -2.293306E-01 -2.751968E+02 1110 LS -1.719980E+03 -1.074987E+03 20 LS -1.156730E-01 -1.388076E+02 1111 LS -8.675476E+02 -5.422172E+02 21 LS -1.648933E-01 -1.978720E+02 1112 LS -1.236700E+03 -7.729375E+02 22 LS -2.805383E-01 -3.366459E+02 1113 LS -2.104037E+03 -1.315023E+03 23 LS -4.403211E-01 -5.283854E+02 1114 LS -3.302408E+03 -2.064005E+03 24 LS -3.472629E-01 -4.167155E+02 1115 LS -2.604472E+03 -1.627795E+03 25 LS -1.096741E-01 -1.316089E+02 1116 LS -8.225557E+02 -5.140973E+02 26 LS -2.777997E-01 -3.333597E+02 1117 LS -2.083498E+03 -1.302186E+03 27 LS -5.641140E-01 -6.769368E+02 1118 LS -4.230854E+03 -2.644284E+03 28 LS -1.514845E+00 -1.817814E+03 1119 LS -1.136134E+04 -7.100837E+03 29 LS -4.686868E-02 -5.624242E+01 1120 LS -3.515151E+02 -2.196969E+02 30 LS -2.175127E-01 -2.610153E+02 1121 LS -1.631346E+03 -1.019591E+03 31 LS -5.620423E-01 -6.744507E+02 1122 LS -4.215317E+03 -2.634573E+03 32 LS -1.723270E+00 -2.067924E+03 1123 LS -1.292452E+04 -8.077828E+03 33 LS 5.910913E-06 7.093096E-03 1124 LS 4.433185E-02 2.770741E-02 34 LS -1.246974E-01 -1.496369E+02 1125 LS -9.352308E+02 -5.845192E+02 35 LS -4.382653E-01 -5.259183E+02 1126 LS -3.286990E+03 -2.054368E+03 36 LS -1.709366E+00 -2.051239E+03 1127 LS -1.282024E+04 -8.012652E+03 37 LS 2.074346E-02 2.489215E+01 1128 LS 1.555759E+02 9.723495E+01 38 LS -4.688111E-02 -5.625734E+01 1129 LS -3.516083E+02 -2.197552E+02 39 LS -1.931140E-01 -2.317368E+02 1130 LS -1.448355E+03 -9.052219E+02 40 LS -1.447031E+00 -1.736438E+03 1131 LS -1.085274E+04 -6.782960E+03 41 LS 3.751652E-02 4.501982E+01 3100 LS 2.813739E+02 1.758587E+02 42 LS 1.016392E-03 1.219671E+00 3101 LS 7.622944E+00 4.764340E+00 43 LS 3.669803E-03 4.403764E+00 3102 LS 2.752353E+01 1.720220E+01 44 LS 1.666951E-02 2.000341E+01 3103 LS 1.250213E+02 7.813833E+01 45 LS 2.865467E-02 3.438560E+01 3104 LS 2.149100E+02 1.343188E+02 46 LS 4.867668E-04 5.841202E-01 3105 LS 3.650751E+00 2.281719E+00 47 LS 3.336146E-03 4.003375E+00 3106 LS 2.502109E+01 1.563818E+01 48 LS 1.955092E-02 2.346111E+01 3107 LS 1.466319E+02 9.164496E+01 49 LS 2.050808E-02 2.460970E+01 3108 LS 1.538106E+02 9.613164E+01 50 LS -8.179694E-03 -9.815634E+00 3109 LS -6.134771E+01 -3.834232E+01 51 LS -5.041794E-04 -6.050153E-01 3110 LS -3.781346E+00 -2.363341E+00 52 LS 1.912964E-02 2.295556E+01 3111 LS 1.434723E+02 8.967017E+01 53 LS 2.040322E-02 2.448386E+01 3112 LS 1.530241E+02 9.564007E+01 54 LS -1.897506E-02 -2.277007E+01 3113 LS -1.423129E+02 -8.894558E+01 55 LS -9.822684E-03 -1.178722E+01 3114 LS -7.367014E+01 -4.604383E+01 56 LS 1.501389E-02 1.801666E+01 3115 LS 1.126041E+02 7.037759E+01 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. 3.155028E-02

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Static Aeroelastic Analysis Sample Problems

ALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING D I S P L A C E M E N T V E C T O POINT ID. TYPE T1 T2 T3 R1 90 G 0.0 0.0 0.0 0.0 97 G 0.0 0.0 0.0 6.577028E-05 98 G 0.0 0.0 0.0 6.577028E-05 99 G -7.856588E-22 -1.774797E-18 0.0 -6.577028E-05 100 G -2.356976E-21 2.024627E-04 0.0 -3.288514E-04 110 G -3.036941E-04 2.712499E-05 1.130680E-02 -8.066081E-04 111 G -3.036941E-04 -1.247221E-04 3.560499E-02 -8.066081E-04 112 G -3.036941E-04 1.789720E-04 -1.299139E-02 -8.066081E-04 120 G -9.110823E-04 -3.235506E-04 9.708962E-02 -1.389569E-03 121 G -9.110823E-04 -4.753977E-04 1.577132E-01 -1.389569E-03 122 G -9.110823E-04 -1.717036E-04 3.646603E-02 -1.389569E-03 310 G -2.356976E-21 1.986964E-03 1.969445E-36 -1.676225E-04 311 G -2.356976E-21 1.162925E-03 -6.460454E-37 -1.676225E-04 312 G -2.356976E-21 2.811003E-03 4.584934E-36 -1.676225E-04 HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING D I S P L A C E M E N T

SUBCASE 1 R R2 0.0 0.0 0.0 0.0 0.0 9.719276E-03 9.719276E-03 9.719276E-03 2.424943E-02 2.424943E-02 2.424943E-02 -1.046196E-36 -1.046196E-36 -1.046196E-36

R3 0.0 0.0 0.0 -6.648222E-19 6.073882E-05 6.073882E-05 6.073882E-05 6.073882E-05 6.073882E-05 6.073882E-05 6.073882E-05 3.296157E-04 3.296157E-04 3.296157E-04

SUBCASE 2 V E C T O R

POINT ID. TYPE T1 T2 T3 R1 R2 R3 90 G 0.0 0.0 0.0 0.0 0.0 0.0 97 G 0.0 0.0 0.0 -4.769856E-06 0.0 0.0 98 G 0.0 0.0 0.0 -4.769856E-06 0.0 0.0 99 G 2.399211E-19 -2.583937E-19 0.0 4.769856E-06 0.0 -9.856156E-20 100 G 7.197632E-19 3.694714E-05 0.0 2.384928E-05 0.0 1.108414E-05 110 G -5.542071E-05 4.950001E-06 6.249369E-03 1.289376E-04 4.107157E-03 1.108414E-05 111 G -5.542071E-05 -2.276035E-05 1.651726E-02 1.289376E-04 4.107157E-03 1.108414E-05 112 G -5.542071E-05 3.266035E-05 -4.018524E-03 1.289376E-04 4.107157E-03 1.108414E-05 120 G -1.662621E-04 -5.904429E-05 5.385996E-02 2.887151E-04 1.153925E-02 1.108414E-05 121 G -1.662621E-04 -8.675465E-05 8.270809E-02 2.887151E-04 1.153925E-02 1.108414E-05 122 G -1.662621E-04 -3.133394E-05 2.501183E-02 2.887151E-04 1.153925E-02 1.108414E-05 310 G 7.197632E-19 -3.824723E-05 9.339349E-35 1.167260E-05 -4.451331E-35 -4.227443E-06 311 G 7.197632E-19 -2.767862E-05 -1.788979E-35 1.167260E-05 -4.451331E-35 -4.227443E-06 312 G 7.197632E-19 -4.881584E-05 2.046768E-34 1.167260E-05 -4.451331E-35 -4.227443E-06 HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING SUBCASE 1 F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 100 0.0 3.456485E-11 0.0 2.287579E-11 0.0 2.337814E-12 -4.525395E-13 -3.551595E+03 101 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.842171E-14 102 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -3.551595E+03 103 0.0 2.284248E-11 0.0 -2.623917E+03 0.0 2.623917E+02 -4.525395E-13 -7.103189E+03 110 1.977727E+05 1.818989E-11 1.872521E+05 5.062471E-12 1.822220E+03 2.273737E-12 3.637979E-12 3.748168E+05 120 1.317888E+05 1.637090E-11 1.609371E+05 1.111993E-11 -2.524316E+03 4.547474E-13 -9.094947E-13 2.787513E+05 310 4.102077E-29 6.507272E+03 4.584849E-30 -3.918364E+03 6.310887E-30 1.805773E+03 9.466331E-30 6.786811E+03 HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING SUBCASE 2 F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 100 0.0 4.879410E-12 0.0 3.636308E-12 0.0 2.486204E-13 1.381945E-10 2.575722E+02 101 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.552714E-15 102 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.575722E+02 103 0.0 3.646647E-12 0.0 -4.788349E+02 0.0 4.788349E+01 1.381945E-10 5.151445E+02 110 9.012888E+04 1.822855E-09 9.559770E+04 1.179611E-09 -9.472266E+02 1.114131E-10 -8.344614E-11 1.517433E+05 120 8.744491E+04 1.153694E-09 7.945616E+04 4.836493E-11 6.918464E+02 9.572432E-11 -4.183676E-11 1.376222E+05 310 1.817536E-27 -1.683244E+03 -5.143632E-28 2.418574E+02 4.038968E-28 -3.334373E+02 1.615587E-27 -4.189096E+02 ALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING S T R E S S E S ELEMENT SA1 SA2 SA3 ID. SB1 SB2 SB3 100 -2.304324E-10 2.304324E-10 -2.304324E-10 -1.525052E-10 1.525052E-10 -1.525052E-10 101 0.0 0.0 0.0 0.0 0.0 0.0 102 0.0 0.0 0.0 0.0 0.0 0.0 103 -1.522832E-10 1.522832E-10 -1.522832E-10 1.749278E+04 -1.749278E+04 1.749278E+04 110 -5.695858E+05 -5.695858E+05 5.695858E+05 -5.392864E+05 -5.392864E+05 5.392864E+05 120 -3.795520E+05 -3.795520E+05 3.795520E+05 -4.634991E+05 -4.634991E+05 4.634991E+05 310 -1.952182E+04 1.952182E+04 -1.952182E+04 1.175509E+04 -1.175509E+04 1.175509E+04

SUBCASE 1 B A R E L E M E N T S ( C B A R ) SA4 AXIAL SA-MAX SA-MIN M.S.-T SB4 STRESS SB-MAX SB-MIN M.S.-C 2.304324E-10 -2.262697E-13 2.302061E-10 -2.306586E-10 1.525052E-10 1.522790E-10 -1.527315E-10 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.522832E-10 -2.262697E-13 1.520569E-10 -1.525095E-10 -1.749278E+04 1.749278E+04 -1.749278E+04 5.695858E+05 2.425319E-12 5.695858E+05 -5.695858E+05 5.392864E+05 5.392864E+05 -5.392864E+05 3.795520E+05 -6.063298E-13 3.795520E+05 -3.795520E+05 4.634991E+05 4.634991E+05 -4.634991E+05 1.952182E+04 1.262177E-29 1.952182E+04 -1.952182E+04 -1.175509E+04 1.175509E+04 -1.175509E+04 I N

HALF-SPAN MODEL, STATIC ANTISYMMETRIC LOADING S T R E S S E S I N B A R E L E M E N T S SA2 SA3 SA4 AXIAL SB2 SB3 SB4 STRESS 3.252940E-11 -3.252940E-11 3.252940E-11 6.909727E-11 2.424206E-11 -2.424206E-11 2.424206E-11 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.431098E-11 -2.431098E-11 2.431098E-11 6.909727E-11 -3.192233E+03 3.192233E+03 -3.192233E+03 -2.595713E+05 2.595713E+05 2.595713E+05 -5.563076E-11 -2.753215E+05 2.753215E+05 2.753215E+05

ELEMENT SA1 ID. SB1 100 -3.252940E-11 -2.424206E-11 101 0.0 0.0 102 0.0 0.0 103 -2.431098E-11 3.192233E+03 110 -2.595713E+05 -2.753215E+05

SUBCASE 2 ( C B A R ) SA-MAX SA-MIN M.S.-T SB-MAX SB-MIN M.S.-C 1.016267E-10 3.656787E-11 9.333933E-11 4.485521E-11 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 9.340825E-11 4.478629E-11 3.192233E+03 -3.192233E+03 2.595713E+05 -2.595713E+05 2.753215E+05 -2.753215E+05

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120 310

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-2.518415E+05 -2.288339E+05 5.049731E+03 -7.255722E+02

-2.518415E+05 -2.288339E+05 -5.049731E+03 7.255722E+02

2.518415E+05 2.288339E+05 5.049731E+03 -7.255722E+02

NX Nastran Aeroelastic Analysis User’s Guide

2.518415E+05 2.288339E+05 -5.049731E+03 7.255722E+02

-2.789117E-11 2.154116E-27

2.518415E+05 2.288339E+05 5.049731E+03 7.255722E+02

-2.518415E+05 -2.288339E+05 -5.049731E+03 -7.255722E+02

Static Aeroelastic Analysis Sample Problems

Listing 6-18. Output for FSW Airplane in Antisymmetric Maneuvers

6.6 FSW Airplane in Unsymmetric Quasi-Steady Maneuvers (Example HA144E) This example demonstrates the use of the NX Nastran static aeroelastic capability to predict the loads produced in unsymmetric maneuvers. A full-span model is necessary to analyze the loads and stresses of an unsymmetric vehicle or of a symmetric vehicle if its maneuvers are unsymmetric. The example here is a full-span model of the FSW airplane considered in symmetric flight in Example HA144A and in antisymmetric maneuvers in Example HA144D. It is only necessary to add the left wing, aileron, and the left side of the canard to the antisymmetric model, which already contains the right side as well as the fin and rudder, and to double the for the stiffnesses and weights fuselage and fin. The left wing is added to the Bulk Data Section beginning with GRIDs 210 through 212 and 220 through 222 which are connected to the left wing elastic axis by RBARs 211, 212, 221, and 222. The left wing elastic axis consists of CBARs 210 and 220 and has the same properties as the right wing. The left wing masses are defined by CONM2s 211, 212, 221, and 222. The left canard structure is described by fuselage degrees of freedom (as is the right canard) and needs no new data. The aerodynamic reference geometry and symmetry are modified on the AEROS entry due to the full-span model: CORD2R 100 is the NACA reference coordinate system; the reference chord is 10.0 ft; the reference span is 40.0 ft; the reference area becomes 400.0 ft2; and SYMXZ = 0 for no symmetry. The left wing aerodynamics are added by CAERO1 2100 with its spline SPLINE2 2601 along its axis on CORD2R 20 and are connected to the left wing grid points using SET1 2100. The left canard aerodynamics are added by CAERO1 2000 along with SPLINE2 1501 and SET1 1000. The left aileron boxes are listed on AELIST 2100, and the total aileron system is given by AESURF 517, which includes both left and right ailerons. The left aileron hinge line is specified on CORD2R 210 and gives a positive aileron deflection with trailing edge down, which is the opposite from the right aileron. The left canard boxes are also added to AESURF 505 using AELlST 2000. CORD2R 1 defines the positive rotation (leading edge up) for both sides since the left and right canard rotate together. The symmetric and antisymmetric controls (AESURF entries) and motions (AESTAT entries) are combined in this example and result in three control rotations: •

canard, δe = ELEV, aileron

•

δa = AILERON

•

rudder, δr , = RUDDER

Ten motions also result: •

angle of attack, α = ANGLEA

•

sideslip angle, β = SIDES

•

pitch rate,

•

roll rate, pb / 2V = ROLL

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•

yaw rate, rb / 2V = YAW

•

vertical load factor,

•

side load factor,

•

pitch acceleration,

•

roll acceleration

•

yaw acceleration,

In anticipation of using this model in later aeroelastic design sensitivity and optimization studies, the model is divided between basic data that will not be involved in the optimization, and design data that will be involved. The basic information is contained in the input file FSW_TWO.DAT (see Listing 6-19), which includes all of the configuration Bulk Data entries for GRIDs, the fuselage structure and weight, and the aerodynamic model. The remaining configuration entries are contained in the Bulk Data Section as shown in Listing 6-20, which includes the right wing, left wing, and fin structure, as well as the weight data.

Symmetrical Cruise One symmetrical and four unsymmetrical flight conditions are demonstrated in this example. The symmetrical condition is included for comparison with Example HA144A . All of the flight conditions are at high speed at sea level with = 1200 psf and a Mach number m = 0.9. With a speed of sound at sea level of 1117 ft/sec, the m = 0.9 airspeed is V = 0.9 × 1117 = 1005 ft/s. For level flight, the vertical load factor is nz = URDD3 = −1.0 in the NACA frame of reference (z is positive downward), the pitch rate is zero, the aileron and rudder rotations are zero, and the five accelerations are all zero. With eight trim variables specified, the remaining five are determined: angle of attack, sideslip angle, canard rotation, and roll and yaw rates. The symmetry of the flight condition should result in computed zeroes for the antisymmetric trim variables.

Steady Rolling Pullout The second flight condition is the first unsymmetrical maneuver. TRIM 2 is a high-speed steady rolling pullout with a vertical load factor of nz = URDD3 = −4.8. Corresponding to the load factor is the steady pitch rate of

so = 2V PITCH = 0.12165 × 0.12165 × 10.0/2 × 1005.0 = 0.000605. The steady yaw rate rb/2V = YAW = 0.0, and the lateral acceleration and roll, pitch, and yaw accelerations are zero: URDD2 = URDD4 = URDD5 = URDD6 = 0.0. The five variables to be determined are angle of attack, canard rotation, sideslip angle, rudder rotation, and roll rate. The second unsymmetrical maneuver is TRIM 3, which is a high-speed pullout with an abrupt roll; therefore, the trim variables are the same as on the TRIM 2 entry except that the roll rate pb / 2V ROLL = 0.0 instead of the roll acceleration being zero. The five variables to be determined are now the angle of attack, canard rotation, sideslip angle, rudder rotation, and acceleration in roll.

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Snap Roll The third unsymmetrical flight condition is TRIM 4, which is a transition from level flight into a snap-roll with maximum rotations of canard and rudder that are assumed at δe = ELEV = 25.0 deg = 0.436332 rad = δr = RUDDER. From SUBCASE 1 the initial angle of attack is α = ANGLEA = 0.0031512 rad = 0.18 deg. The variables that are assumed to be zero at the beginning of the maneuver are aileron rotation, sideslip angle, and pitch, roll, and yaw rates. The five resulting accelerations will be found.

Steady Climbing Turn The last unsymmetrical maneuver, TRIM 5, is a steady climbing turn. This type of maneuver is thoroughly analyzed by Etkin (1972, pp. 423-428). A climb rate of R/C = 1500.00 ft/min and a horizontal turning rate of ω = 3.0 deg/sec are assumed along with the Mach number m = 0.9 at sea level assumed in the previous subcases. The turn coordinator gyro instrument actually measures the yaw rate r but is calibrated to give the standard turning rate of 3.0 deg/sec at the bank angle that would be flown in a level turn at cruise speed. The angular rates in a steady turn with a small climb angle θ as given in Etkin [1972, Eqs. (10.4,5)] are

and the bank angle φ is given by Etkin [1972, Eqs. (10.4,7)] as

For this subcase ω = 3.0 deg/sec = 0.052360 rad/sec; therefore, the bank angle becomes

or φ = 58.56 deg. The pitch and yaw rates become

The roll rate requires the pitch angle in the climb, which is found from the vertical speed and the angle of attack. The angle of attack in the climb can be estimated from the load factor [Etkin (1972, Eqs. (10.4,9))].

and the level flight trim angle of attack from SUBCASE 1 of α = 0.003152 to be

The angle of climb is

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Thus, the pitch angle is

Finally, the roll rate becomes

The dimensionless angular rates then become

The remaining input parameters are the zero accelerations: URDD2 = URDD4 = URDD5 = URDD6 = 0.0. The five trim variables to be determined are the angle of attack, the canard rotation, the sideslip angle, the aileron rotation, and the rudder rotation. The ENDDATA entry completes the Bulk Data Section. The trim data for the five flight conditions are also shown in Listing 6-20.

Case Control Commands The first three entries of the Case Control Section list the headers that appear at the top of every page of output and at the bottom of every plot output. The ECHO = BOTH command specifies that both the unsorted and sorted Bulk Data entries be printed in the output. The SPC command is required by the corresponding Bulk Data entries. The DlSP command specifies that all displacements be printed. The STRESS, FORCE, AEROF, and APRES commands request that all stress, force, aerodynamic force, and pressure data be printed. The pairs of SUBCASE and TRIM commands are also required by the corresponding Bulk Data entries; the sets of output data are identified according to the subcase number, which may be different from the TRIM command ID. The remaining commands in the Case Control Section specify output plot parameters. The OUTPUT(PLOT) command delimits the output request designated by its argument, in this case, the structure plotter. The PLOTTER NASTRAN command invokes the NASTRAN plotter routine. The SET 1 command specifies that all deflections be plotted. The FIND command requests that the computer program determine the scale and origin for the set to be plotted. Then the PLOT command invokes the plot routine and specifies that the static deformations be plotted and outlined. The BEGIN BULK command completes the Case Control Section. The first statement in the Executive Control Section is ID NXN, HA144E, which is the identification of this example. The next statement TIME 5 restricts the CPU time to 5.0 minutes of computing. SOL 144 calls for the Static Aeroelastic Response DMAP sequence. Finally, the CEND statement completes the Executive Control Section.

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Output The Executive Control and Case Control data are shown in Listing 6-20. The sorted Bulk Data entries are presented in Listing 6-21, and the output results follow in Listing 6-22. The output data to be discussed begin with the OUTPUT FROM GRID POINT WEIGHT GENERATOR, which gives the inertial characteristics of the full-span model, including its gross weight of 16,100 lbs. The next output table is the NON-DIMENSIONAL STABILITY AND CONTROL DERIVATIVES. The coefficients for longitudinal and lateral-directional motions are preceded by the transformation from the basic to the reference coordinates; then the rigid and elastic coefficients are tabulated. The rigid coefficients include those calculated with and without the spline as a check on the splining. The elastic coefficients are tabulated for both the restrained and unrestrained conditions. The rigid coefficients can be compared with the longitudinal coefficients of Example HA144A, and the lateral-directional coefficients of Example HA144D, and they agree exactly. The elastic coefficients can also be compared with the same previous examples: the longitudinal restrained aerodynamic coefficients agree exactly, but the inertial coefficients differ slightly because Example HA144A does not include the fin weight. The lateral-directional coefficients agree very closely because Example HA144D has the same gross weight. The INTERMEDIATE MATRIX... HP follows with the mean axis deflections and rotations. Note that the printed format is the transpose of the matrix. The first, third, and fifth rows of values shown beginning with the sixth column can be compared with the three rows given in the lateral-directional case of Example HA144D; the second and fourth rows of values shown beginning in the first column can be compared closely (because of the different gross weights) with the two rows and five columns given in the Iongitudinal case of Example HA144A. The remaining output includes the AEROSTATIC DATA RECOVERY OUTPUT TABLE which contains the trim solutions (AEROELASTIC TRIM VARIABLES) and aerodynamic pressure coefficients, pressures, and loads (NORMAL FORCES and MOMENTS on each aerodynamic box). The aeroelastic coefficients are output for each subcase but are only shown in Listing 6-22 for the first subcase. The standard NX Nastran output of displacements, element forces, and element stresses follows the aeroelastic output. SUBCASE 1 determines the trim variables in level flight and is followed by the aerodynamic loads. The angle of attack is α = 0.003151 rad = 0.18 deg and canard incidence is δe = 0.01725 rad = 0.99 deg. The sideslip angle and the yaw and roll rates are computed zeroes. The SUBCASE 2 trim solution and aerodynamic loads for the rolling pullout are shown next. The pullout at 4.8 g’s results in an angle of attack α = 0.01370 rad = 0.78 deg and canard incidence δe = 0.08540 rad = 4.89 deg. The steady roll rate achieved with the aileron deflection δa of 25 deg is pb/2V = 0.2600 and is accompanied by a sideslip angle β = −0.01824 rad = −1.05 deg and rudder rotation δr = 0.02059 rad = 1.18 deg. The positive rudder rotation is required because the lifting surface theory (Vortex-Lattice method) predicts a proverse yawing moment from the ailerons. The SUBCASE 3 trim solution and aerodynamic loads are then shown for the maneuver of a pullout with an abrupt roll. The pullout results in the same angle of attack and canard incidence or as before, but the abrupt aileron input now results in a roll acceleration with a sideslip angle β = −0.06297 rad = −3.61 deg and rudder rotation δr = 0.01100 rad = 0.63 deg. The SUBCASE 4 trim solution and aerodynamic loads are shown next for the snap-roll entry. The resulting accelerations are a vertical load factor of nz = −8.660 g’s, a lateral load factor of ny = 3.127 g’s, a pitching acceleration of

or

, a rolling acceleration

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Chapter 6

of

or

, and a yawing acceleration of

or

. The SUBCASE 5 trim solution and aerodynamic loads for the climbing turn are shown last. The angle of attack is α = 0.005519 rad = 0.32 deg, and the canard angle is δe = 0.03402 rad = 1.94 deg. The input data for this case required a pitch angle to determine the roll rate. The pitch angle was estimated from the sum of the climb angle, known from the vertical speed, and the angle of attack, which was estimated from the load factor in the turn and the previously determined angle of attack in level flight. The input roll rate was based on a pitch angle calculation that assumed an angle of attack of 0.006042 rad = 0.35 deg. The assumed angle of attack is close to the value just obtained from the climbing turn solution. Another iteration could be made to bring these numbers into even closer agreement, but it does not appear necessary. Only small control surface rotations and sideslip angle are required to coordinate the example turn: aileron, δa = −0.00005857 rad = − 0.0034 deg, rudder, δr = −0.0001941 rad = − 0.011 deg, and sideslip angle, β = 0.0004591 rad = 0.026 deg.

Data Recovery Following the SUBCASE 5 trim solution and aerodynamic loads are the outputs for the five subcases for displacements, forces in the BAR elements, and the stresses in the BAR elements. The stresses may appear to be high, but recall that their units are psf. The differences between the stresses in the right and left sides are of interest in the unsymmetrical maneuvers. The higher levels will, of course, be the critical ones for the purpose of aircraft design. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ * * * STRUCTURAL DATA * * * $ $ $ $ (LB-FT-SEC SYSTEM) $ $ $ $ * * GRID GEOMETRY * * $ $ $ $ GRID 90 - 100 (T3) FUSELAGE POINTS $ $ GRID 110 - 122 (T3) WING POINTS $ $ GRID 310 - 312 (t3) FIN POINTS $ $ $ $ * FUSELAGE GRID * $ $ $ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $ $ ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS AND $ $ ITS ASSOCIATED SUPERELEMENT ID. $ $ $ $ ID CP X1 X2 X3 CD PS SEID GRID 90 15. 0. 0. GRID 97 0. 0. 0. GRID 98 10. 0. 0. GRID 99 20. 0. 0. GRID 100 30. 0. 0. $ $ $ * RIGHT WING GRID * $ $ $ $ ID CP X1 X2 X3 CD PS SEID GRID 111 24.61325 +5. 0. GRID 110 27.11325 +5. 0. GRID 112 29.61325 +5. 0. GRID 121 18.83975+15. 0. GRID 120 21.33975+15. 0. GRID 122 23.83975+15. 0. $ $ $ * LEFT WING GRID * $ $ $ $ ID CP X1 X2 X3 CD PS SEID GRID 211 24.61325 -5. 0. GRID 210 27.11325 -5. 0.

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GRID GRID GRID GRID $ $ $ GRID GRID GRID $ $ $ $ $ $ $ $ $ $ $ $ CBAR CBAR CBAR CBAR $ $ $ $ $ $ $ $ $ $ $ $ $ PBAR $ +PB1F $ +PB2F $ $ $ $ $ $ $MAT1 MAT1 $ $ $ $ $ $ $ $ $ $ $ $ CONM2 CONM2 CONM2 CONM2 $ $ $ $ $ $ $ $ $ PARAM $ $ $ $ $ $ $ PARAM $ $ $ $ PARAM

212 221 220 222

29.61325 -5. 18.83975-15. 21.33975-15. 23.83975-15.

0. 0. 0. 0. $ $ $

* FIN GRID * 310 311 312

32.88675+0. 30.38675+0. 35.38675+0.

5. 5. 5. $ $ $ $ $ $ $ $ $ $ $

* * STRUCTURAL STIFFNESS PROPERTIES * * * FUSELAGE STRUCTURE * THE CBAR ENTRY DEFINES A SIMPLE BEAM ELEMENT. LISTED ARE ITS PROPERTY ENTRY ID, THE TWO GRID POINTS JOINED BY THE BEAM AND COMPONENTS OF A VECTOR FROM THE FIRST POINT. THIS VECTOR DEFINES THE DIRECTION OF THE STRUCTURAL DEFLECTION OF THE POINT AND ITS POSITIVE SENSE. EID 101 102 103 104

PID 100 100 100 100

GA 97 98 90 99

GB 98 90 99 100

X1,GO 0. 0. 0. 0.

X2 0. 0. 0. 0.

X3 1. 1. 1. 1.

THE PBAR ENTRY DEFINES GEOMETRIC PROPERTIES OF THE BEAM. LISTED ARE ITS ASSOCIATED MATERIAL ENTRY ID, ITS CROSS SECTIONAL AREA, AREA MOMENTS OF INERTIA, TORSIONAL MOMENT OF INERTIA AND NON-STRUCTURAL MASS PER UNIT AREA. THE OPTIONAL CONTINUATION ENTRY CONTAINS STRESS RECOVERY COEFFICIENTS, I.E., Y,Z COORDINATES WHERE STRESSES ARE TO BE COMPUTED. K1 AND K2 ARE AREA FACTORS FOR SHEAR STIFFNESS (DEFAULT IS BLANK; THEN SHEAR STIFFNESS IS INFINITE, I.E., SHEAR FLEXIBILITY IS ZERO. I12 IS THE AREA PRODUCT OF INERTIA. PID 100 C1 1.0 K1

MID

A 4.0 D1 1.0 I12 0.0

1 C2 1.0 K2

I1 .347222 D2 -1.0

I2 .30 E1 -1.0

J 1.0 E2 1.0

NSM +PB1F F1 -1.0

F2 -1.0

THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES. LISTED ARE ITS ID, ITS ELASTIC MODULUS, SHEAR MODULUS, POISSONS RATIO, MASS DENSITY, TEMPERATURE EXPANSION COEFFICIENT, REFERENCE TEMPERATURE AND A STRUCTURAL DAMPING COEFFICIENT. MID 1

E 1.44+9

G NU 5.40+8

RHO 0.0

A

TREF

GE

* FUSELAGE MASSES * THE CONM2 ENTRY DEFINES A CONCENTRATED MASS. LISTED ARE ITS ID, GRID LOCATION, COORDINATE SYSTEM TO LOCATE THE CENTER OF GRAVITY, THE MASS VALUE AND THE LOCATION OF THE CENTER OF GRAVITY RELATIVE TO THE GRID LOCATION. G 97 98 99 100

CID 0 0 0 0

M 3000.0 3000.0 3000.0 3000.0

X1

X2

THE PARAM,GRDPNT,XX ENTRY CAUSES THE GRID POINT WEIGHT GENERATOR TO BE EXECUTED USING GRID POINT XX AS THE REFERENCE POINT. THEN THE INERTIA MATRIX, THE TRANSFER MATRIX FROM BASIC TO PRINCIPAL AXES AND OTHER PERTINENT INERTIA DATA ARE PRINTED.

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

90

THE PARAM,WTMASS,GINV CAUSES ALL THE STRUCTURAL MASSES AND MASS DENSITIES TO BE MULTIPLIED BY GINV (I.E., BY ONE OVER THE ACCELERATION OF GRAVITY). THE DYNAMIC PRESSURE SUPPLIED FOR AERODYNAMIC FORCE CALCULATIONS WILL NOT BE MULTIPLIED BY GINV. WTMASS

$ $ $ $ $ $

X3

* * STRUCTURAL PARAMETERS * *

GRDPNT

+PB2F

+MT

* * MASS AND INERTIA PROPERTIES * *

EID 97 98 99 100

$ $ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $

.031081

THE PARAM,AUNITS,GINV PERMITS THE ACCELERATIONS ON THE TRIM ENTRY TO BE SPECIFIED IN UNITS OF LOAD FACTOR (I.E., IN G’S) AUNITS

.031081

NX Nastran Aeroelastic Analysis User’s Guide

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Static Aeroelastic Analysis Sample Problems

Chapter 6

$ $ $ $ $ $ $ SPC1 $ $ $ $ $ $ $ $

* * STRUCTURAL CONSTRAINTS * * THE SPC1 ENTRY CONSTRAINS THE LISTED GRID POINTS IN THE SPECIFIED DOF COMPONENTS. SID 1

C 1

G1 90

G2

G3

G4

THE SUPORT ENTRY IDENTIFIES A GRID POINT OR A SCALAR POINT AND SPECIFIES THE DOF COMPONENTS IN WHICH THE USER DESIRES REACTIONS TO BE APPLIED TO PREVENT RIGID BODY MOTION. IT THUS INVOKES THE SOLUTION OF THE BALANCE EQUATIONS TO DETERMINE THE REACTIONS. IN THE STATIC AEROELASTIC SOLUTION THE DOF COMPONENTS MUST BE CONSISTENT WITH THE UNDEFINED VARIABLES ON THE TRIM ENTRIES

$ $ ID C SUPORT 90 23456 $ $ THE OMIT1 ENTRY IDENTIFIES GRID POINTS TO BE OMITTED FROM $ THE REMAINDER OF THE ANALYSIS. $ $ ID G G G G OMIT1 4 110 120 210 220 310 $ $ * * * AERODYNAMIC DATA * * * $ $ (LB-FT-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AEROS ENTRY IS UNIQUE TO THE STATIC AEROELASTICITY $ SOLUTION, SOL144. ACSID IDENTIFIES THE AERO COORDINATE $ SYSTEM, RCSID IDENTIFIES THE REFERENCE COORDINATE SYS$ TEM FOR RIGID BODY MOTION. REFC IS THE REFERENCE CHORD. $ REFB IS THE REFERENCE SPAN. REFS IS THE REFERENCE WING $ AREA. SYMXZ AND SYMXY ARE SYMMETRY KEYS. $ $ ACSID RCSID REFC REFB REFS SYMXZ SYMXY AEROS 1 100 10.0 40.0 400.0 $ $ THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM $ FLAGGED BY THE AEROS ENTRY. THE ORIGIN IS AT THE CANARD $ QUARTER CHORD. LISTED ARE THE ORIGIN, A POINT ALONG THE $ Z AXIS AND A POINT IN THE X-Z PLANE, ALL IN THE RID $ COORDINATE SYSTEM. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0 12.5 0. 0. 12.5 0. 10. $ C1 C2 C3 +CRD1 20. 0. 0. $ $ THIS CORD2R ENTRY DEFINES THE NACA COORDINATE SYSTEM TO $ WHICH ALL THE STABILITY DERIVATIVES AND TRIM CONDITIONS $ WILL BE REFERENCED. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 -10.0 $ C1 C2 C3 +CRD100 0.0 0.0 0.0 $ $ $ * WING AERODYNAMIC MODEL * $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AERODYNAMICS. $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS, $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ EXTRA POINT IDS. $ $ * RIGHT WING * $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 1100 1000 8 4 1 $ ( FWD LEFT POINT ) CHORD ( FWD RIGHT POINT ) CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CARW 25. 0. 0. 10. 13.45299+20. 0. 10 $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6

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NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CRD1 $ $ $ $ $ +CRD100 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CARW . $ $ $ $

Static Aeroelastic Analysis Sample Problems

PAERO1 1000 $ $ * LEFT WING * CAERO1 2100 1000 8 4 1 +CALW 13.45299-20. 0. 10. 25. 0. 0. 10. $ $ * CANARD AERODYNAMIC MODEL * $ $ * RIGHT SIDE * CAERO1 1000 1000 2 4 1 +CARC 10. 0. 0. 10. 10. 5. 0. 10. $ $ * LEFT SIDE * CAERO1 2000 1000 2 4 1 +CALC 10. -5. 0. 10. 10. 0. 0. 10. $ $ * FIN AERODYNAMIC MODEL * $ CAERO1 3100 1000 4 4 1 +CA1FI 30.7735 0. 10 10. 25. 0. 0. 10. $ $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * BEAM SPLINE FIT ON THE WING * $ $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT$ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR $ ATTACHMENT AND TORSIONAL FLEXIBILITIES. CID IDENTIFIES $ THE CORD2R ENTRY THAT DEFINES THE SPLINE AXIS. DTHX AND $ DTHY ARE ROTATIONAL ATTACHMENT FLEXIBILITIES (-1. SPECIFIES $ NO ATTACHMENT). $ $ * RIGHT WING * $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 1601 1100 1100 1131 1100 0. 1. 2 $ DTHX DTHY +SPRW -1. -1. $ $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 SET1 1100 99 100 111 112 121 122 $ $ * LEFT WING * $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 2601 2100 2100 2131 2100 0. 1. 20 $ DTHX DTHY +SPLW -1. -1. $ $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 SET1 2100 99 100 211 212 221 222 $ $ * BEAM SPLINE FIT ON THE CANARD * $ $ * RIGHT SIDE * SPLINE2 1501 1000 1000 1007 1000 0. 1. 1 +SPRC 1. -1. $ SET1 1000 98 99 $ $ * LEFT SIDE * SPLINE2 2501 2000 2000 2007 1000 0. 1. 1 +SPLC 1. -1. $ $ * BEAM SPLINE FIT ON THE FIN * $ SPLINE2 3100 3100 3100 3115 3100 0. 1. 300 +SP2FI -1. -1. $ SET1 3100 99 100 311 312 $ $ THE CORD2R ENTRY DEFINES THE COORDINATE SYSTEM IN WHICH THE $ BEAM SPLINE EXTENDS ALONG THE WING Y-AXIS. IT LISTS THE $ ORIGIN, A POINT ALONG THE Z-AXIS AND A POINT IN THE X-Z $ PLANE. $ * RIGHT WING SPLINE AXIS * $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 2 0 30. 0. 0. 30. 0. 10. $ C1 C2 C3 +CRD2RW 38.66025+5.0 0. $ $ * LEFT WING SPLINE AXIS * $

$ +CALW $ $ $ +CARC $ +CALC $ $ $ +CA1FI $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +SPRW $ $ $ $ $ $ +SPLW $ $ $ $ $ $ $ $ +SPRC $ $ $ +SPLC $ $ $ +SP2FI $ $ $ $ $ $ $ $ +CRD2RW $ $ $

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 6

Static Aeroelastic Analysis Sample Problems

$ CID CS A1 A2 A3 B1 B2 B3 CORD2R 20 0 30. 0. 0. 30. 0. 10. $ C1 C2 C3 +CRD2LW 38.66025-5.0 0. $ $ * FIN SPLINE AXIS * $ CORD2R 300 0 30.0 0. 0. 30.0 10.0 0. +CRD2FI 20.0 0.0 5.7735 $ $ * CONTROL SURFACE DEFINITION * $ $ THE AESURF ENTRY DEFINES AN AERODYNAMIC CONTROL SURFACE. $ LISTED ARE THE ALPHANUMERIC NAME OF THE SURFACE, THE ID $ OF A COORDINATE SYSTEM THAT DEFINES THE HINGE LINE AND $ THE ID OF AN AELIST ENTRY. $ $ ID LABEL CID1 ALID1 CID2 ALID2 AESURF 505 ELEV 1 1000 1 2000 AESURF 517 AILERON 110 1100 210 2100 AESURF 518 RUDDER 301 3000 $ $ THE AELIST ENTRY LISTS AERODYNAMIC BOXES THAT LIE ON THE $ CONTROL SURFACE. $ $ SID E1 E2 E3 ETC AELIST 1000 1000 THRU 1007 AELIST 2000 2000 THRU 2007 AELIST 1100 1119 1123 1127 1131 AELIST 2100 2103 2107 2111 2115 AELIST 3000 3103 3107 3111 3115 $ $ * CONTROL SURFACE HINGE LINES * $ $ * CANARD * $ THE COORDINATE SYSTEM, CORD2R,1, REFERENCED BY THE AEROS ENTRY $ IS THE CANARD HINGE LINE, AND NEEDS NO FURTHER DEFINITION $ $ * RIGHT AILERON * CORD2R 110 0 26.7265 10.0 0. 26.7265 10.0 -10. +CRD2RA 36.7265 15.7735 0. $ $ * LEFT AILERON * CORD2R 210 0 26.7265 -10.0 0. 26.7265 -10.0 10.0 +CRD2LA 36.7265 -15.7735 0. $ $ * RUDDER * CORD2R 301 0 32.5 0. 0. 32.5 -10. 0.0 +CRD2R 22.5 0. 5.7735 $ $ * * AERODYNAMIC DOFS * * $ $ THE AESTAT ENTRY LISTS TRIM VARIABLES USED TO SPECIFY $ RIGID BODY MOTIONS. THESE AND THE CONTROL SURFACE $ ROTATIONS MAKE UP THE VARIABLES IN THE EQUATIONS OF $ MOTION. $ $ ID LABEL AESTAT 501 ANGLEA AESTAT 502 PITCH AESTAT 503 URDD3 AESTAT 504 URDD5 AESTAT 511 SIDES AESTAT 512 YAW AESTAT 513 ROLL AESTAT 514 URDD2 AESTAT 515 URDD4 AESTAT 516 URDD6

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NX Nastran Aeroelastic Analysis User’s Guide

+CRD2LW $ $ $ +CRD2FI $ $ $ $ $ $ $ $

$ $ $ $

$ $ $ $ $ $ $ $ +CRD2RA $ $ +CRD2LA $ $ +CRD2R $ $ $ $ $ $ $ $

Static Aeroelastic Analysis Sample Problems

Listing 6-19. FSW_TWO DAT Input File N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA144E $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA144E $$$$$$$$ $ $ $ MODEL DESCRIPTION FULL SPAN 30 DEG FWD SWEPT WING $ $ WITH AILERON, CANARD AND AFT SWEPT $ $ VERTICAL FIN AND RUDDER. $ $ BAR MODEL WITH DUMBBELL MASSES. $ $ $ $ SOLUTION QUASI-STEADY AEROELASTIC ANALYSIS $ $ USING DOUBLET-LATTICE METHOD $ $ AERODYNAMICS AT MACH NO. 0.9. $ $ $ $ OUTPUT PLOTS OF THE STICK MODEL AND AERO $ $ GRID, LISTS OF RESTRAINED AND $ $ UNRESTRAINED SYMMETRIC AND ANTI$ $ SYMMETRIC STABILITY DERIVATIVES, $ $ AERODYNAMIC FORCES AND PRESSURES $ $ PLUS STRESSES AND DEFLECTIONS FOR $ $ LEVEL FLIGHT AND SEVERAL UNSYM$ $ METRICAL MANUEVERS. $ $ $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO CEND EXAMPLE HA144E: 30 DEG FWD SWEPT WING WITH 3 CONTROLS PAGE UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144E: 30 DEG FWD SWEPT WING WITH 3 CONTROLS 2 SUBTI = UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO 3 LABEL = FULL-SPAN MODEL 4 ECHO = BOTH 5 SPC = 1 $ SYMMETRIC CONSTRAINTS 6 SET 1 = 1 THRU 999999 7 DISP = 1 $ PRINT ALL DISPLACEMENTS 8 STRESS = ALL $ PRINT ALL STRESSES 9 FORCE = ALL $ PRINT ALL FORCES 10 AEROF = ALL $ PRINT ALL AERODYNAMIC FORCES 11 APRES = ALL $ PRINT ALL AERODYNAMIC PRESSURES 12 SUBCASE 1 13 TRIM = 1 $ HIGH SPEED LEVEL FLIGHT 14 SUBCASE 2 15 TRIM = 2 $ HIGH SPEED ROLLING PULLOUT 16 SUBCASE 3 17 TRIM = 3 $ HIGH SPEED PULLOUT WITH ABRUPT ROLL 18 SUBCASE 4 19 TRIM = 4 $ HIGH SPEED SNAP-ROLL ENTRY 20 SUBCASE 5 21 TRIM = 5 $ HIGH SPEED CLIMBING TURN 22 OUTPUT(PLOT) 23 PLOTTER = NASTRAN 24 SET 1 = ALL 25 FIND SCALE, ORIGIN 1,SET 1 26 PLOT SET 1 27 PLOT STATIC DEFORMATION 0, ORIGIN 1, SET 1, OUTLINE 28 BEGIN BULK EXAMPLE HA144E: 30 DEG FWD SWEPT WING WITH 3 CONTROLS PAGE 3 UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL I N P U T B U L K D A T A D E C K E C H O

2

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ INCLUDE FSW_TWO.DAT $ $ $ $ * RIGHT WING STRUCTURE * $ $ $ $ EID PID GA GB X1,GO X2 X3 CBAR 110 101 100 110 0. 0. 1. CBAR 120 101 110 120 0. 0. 1. $ $ $ THE RBAR ENTRY DEFINES A RIGID BAR. LISTED ARE THE GRID $ $ POINTS AT EACH END AND THE DEPENDENT AND INDEPENDENT DOFS $ $ AT EACH END. THE NUMBER OF INDEPENDENT DOFS AT THE TWO $ $ ENDS MUST EQUAL SIX. BY DEFAULT THOSE NOT DECLARED INDE$ $ PENDENT ARE MADE DEPENDENT. NX Nastran Aeroelastic Analysis User’s Guide $ 6-85 $ $ $ EID GA GB CNA CNB CMA CMB

Chapter 6

Static Aeroelastic Analysis Sample Problems

$ $ PBAR $ +PB1W $ +PB2W $ $ $ $ $ CBAR CBAR $ RBAR RBAR RBAR RBAR $ $ $ CBAR $ RBAR RBAR $ $ $ $ $ $ $ MAT1 $ $ $ CONM2 CONM2 CONM2 CONM2

$ PID 101 C1 0.5 K1

MID 2 C2 3.0 K2

A 1.5 D1 0.5 I12 0.0

I1 I2 0.173611+2.0 D2 E1 -3.0 -0.5

+PB1W F2 -3.0

+PB2W $

* LEFT WING STRUCTURE * EID 210 220

PID 101 101

GA 100 210

GB 210 220

211 212 221 222

210 210 220 220

211 212 221 222

123456 123456 123456 123456

X1,GO 0. 0.

$ $ X2 0. 0.

X3 1. 1.

$ $ $

* FIN STRUCTURE * 310

101

100

310

311 312

310 310

311 312

123456 123456

0.

0.

1. $

THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES. LISTED ARE ITS ID, ITS ELASTIC MODULUS, SHEAR MODULUS, POISSONS RATIO, MASS DENSITY, TEMPERATURE EXPANSION COEFFICIENT, REFERENCE TEMPERATURE AND A STRUCTURAL DAMPING COEFFICIENT. MID 2

E 1.44+9

G 5.40+8

NU

RHO

A

TREF

111 112 121 122

111 112 121 122

0 0 0 0

$ $ $ $ $ $

GE $ $ $

* RIGHT WING MASSES * 600.0 400.0 600.0 400.0

$

$ $ $ CONM2 CONM2 CONM2 CONM2 $ $ $ CONM2 CONM2 $ $ $ $ $ $ $ $ $ $ $ $ $ TRIM +TR1A +TR1B $ $ $ TRIM +TR2A +TR2B $ $ $ TRIM +TR3A +TR3B $ $ $ TRIM +TR4A +TR4B $ $ $ TRIM

6-86

J NSM 0.462963 E2 F1 3.0 -0.5

* LEFT WING MASSES * 211 212 221 222

211 212 221 222

0 0 0 0

$ $

600.0 400.0 600.0 400.0 $ $ $

* FIN MASSES * 311 312

311 312

0 0

60.0 40.0

0.

$ $ $ $ $ $ $ $ $ $ $ $ $ +TR1A +TR1B

0.

$ $ $ +TR2A +TR2B

0.

$ $ $ +TR3A +TR3B

.436332 YAW 0.

$ $ $ +TR4A +TR4B

-1.9170

$ $ $ +TR5A

* * TRIM CONDITIONS * * THE TRIM ENTRY SPECIFIES CONSTRAINTS FOR THE TRIM VARIABLES LISTED ON THE AESTAT AND AESURF ENTRIES. LISTED ARE ITS ID, THE MACH NUMBER, DYNAMIC PRESSURE AND PAIRS OF TRIM VARIABLES AND THEIR CONSTRAINED VALUES. THOSE THAT ARE NOT HELD FIXED MUST BE CONSTRAINED BY REACTION FORCES STIPULATED ON THE SUPORT ENTRY. SEE SECTION 3.5.3 OF THE THEORETICAL MANUAL FOR MORE DETAILS. TRIM CONDITION 1: LEVEL FLIGHT AT HIGH DYNAMIC PRESSURE 1 URDD5 URDD4

0.9 0. 0.

1200.0 PITCH AILERON 0. URDD6 0.

0. RUDDER

URDD3 0.

-1.0 URDD2

TRIM CONDITION 2: HIGH SPEED ROLLING PULLOUT 2 URDD5 URDD4

0.9 0. 0.

1200.0 PITCH 6.0499-4URDD3 AILERON .436332 YAW 0. URDD6 0.

-4.8 URDD2

TRIM CONDITION 3: HIGH SPEED PULLUP WITH ABRUPT ROLL 3 URDD5 URDD2

0.9 0. 0.

1200.0 PITCH 6.0499-4URDD3 AILERON .436332 ROLL 0. URDD6 0.

-4.8 YAW

TRIM CONDITION 4: HIGH SPEED ENTRY INTO SNAP-ROLL 4 0.9 PITCH 0. AILERON 0.

1200.0 SIDES RUDDER

ANGLEA .0031512ELEV 0. ROLL 0. .436332

TRIM CONDITION 5: HIGH SPEED CLIMBING TURN 5

0.9

NX Nastran Aeroelastic Analysis User’s Guide

1200.0

PITCH

.2222-3 URDD3

Static Aeroelastic Analysis Sample Problems

+TR5A URDD5 0. YAW +TR5B URDD4 0. URDD6 $ $ ENDDATA INPUT BULK DATA CARD COUNT =

.5435-3 ROLL 0.

-.3222-4URDD2

0.

* * *

+TR5B $ $

502

Listing 6-20. Input Files for FSW Airplane in Unsymmetric Maneuvers EXAMPLE HA144E: 30 DEG FWD SWEPT WING WITH 3 CONTROLS PAGE 14 UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AELIST 1000 1000 THRU 1007 2AELIST 1100 1119 1123 1127 1131 3AELIST 2000 2000 THRU 2007 4AELIST 2100 2103 2107 2111 2115 5AELIST 3000 3103 3107 3111 3115 6AEROS 1 100 10.0 40.0 400.0 7AESTAT 501 ANGLEA 8AESTAT 502 PITCH 9AESTAT 503 URDD3 10AESTAT 504 URDD5 11AESTAT 511 SIDES 12AESTAT 512 YAW 13AESTAT 513 ROLL 14AESTAT 514 URDD2 15AESTAT 515 URDD4 16AESTAT 516 URDD6 17AESURF 505 ELEV 1 1000 1 2000 18AESURF 517 AILERON 110 1100 210 2100 19AESURF 518 RUDDER 301 3000 20CAERO1 1000 1000 2 4 1 +CARC 21+CARC 10. 0. 0. 10. 10. 5. 0. 10. 22CAERO1 1100 1000 8 4 1 +CARW 23+CARW 25. 0. 0. 10. 13.45299+20. 0. 10. 24CAERO1 2000 1000 2 4 1 +CALC 25+CALC 10. -5. 0. 10. 10. 0. 0. 10. 26CAERO1 2100 1000 8 4 1 +CALW 27+CALW 13.45299-20. 0. 10. 25. 0. 0. 10. 28CAERO1 3100 1000 4 4 1 +CA1FI 29+CA1FI 30.7735 0. 10. 10. 25. 0. 0. 10. 30CBAR 101 100 97 98 0. 0. 1. 31CBAR 102 100 98 90 0. 0. 1. 32CBAR 103 100 90 99 0. 0. 1. 33CBAR 104 100 99 100 0. 0. 1. 34CBAR 110 101 100 110 0. 0. 1. 35CBAR 120 101 110 120 0. 0. 1. 36CBAR 210 101 100 210 0. 0. 1. 37CBAR 220 101 210 220 0. 0. 1. 38CBAR 310 101 100 310 0. 0. 1. 39CONM2 97 97 0 3000.0 40CONM2 98 98 0 3000.0 41CONM2 99 99 0 3000.0 42CONM2 100 100 0 3000.0 43CONM2 111 111 0 600.0 44CONM2 112 112 0 400.0 45CONM2 121 121 0 600.0 46CONM2 122 122 0 400.0 47CONM2 211 211 0 600.0 48CONM2 212 212 0 400.0 49CONM2 221 221 0 600.0 50CONM2 222 222 0 400.0 51CONM2 311 311 0 60.0 52CONM2 312 312 0 40.0 53CORD2R 1 0 12.5 0. 0. 12.5 0. 10. +CRD1 54+CRD1 20. 0. 0. 55CORD2R 2 0 30. 0. 0. 30. 0. 10. +CRD2RW 56+CRD2RW 38.66025+5.0 0. 57CORD2R 20 0 30. 0. 0. 30. 0. 10. +CRD2LW 58+CRD2LW 38.66025-5.0 0. 59CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 -10.0 +CRD100 60+CRD100 0.0 0.0 0.0 61CORD2R 110 0 26.7265 10.0 0. 26.7265 10.0 -10.0 +CRD2RA 62+CRD2RA 36.7265 15.7735 0. 63CORD2R 210 0 26.7265 -10.0 0. 26.7265 -10.0 10.0 +CRD2LA 64+CRD2LA 36.7265 -15.77350. 65CORD2R 300 0 30.0 0. 0. 30.0 10.0 0. +CRD2FI 66+CRD2FI 20.0 0.0 5.7735 67CORD2R 301 0 32.5 0. 0. 32.5 -10. 0.0 +CRD2R 68+CRD2R 22.5 0. 5.7735 69GRID 90 15. 0. 0. 70GRID 97 0. 0. 0. 71GRID 98 10. 0. 0. 72GRID 99 20. 0. 0. 73GRID 100 30. 0. 0. 74GRID 110 27.11325+5. 0. 75GRID 111 24.61325+5. 0. 76GRID 112 29.61325+5. 0. 77GRID 120 21.33975+15. 0. 78GRID 121 18.83975+15. 0. 6-87 NX Nastran Aeroelastic Analysis User’s Guide

Static Aeroelastic Analysis Sample Problems

Chapter 6

798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142-

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID MAT1 MAT1 OMIT1 PAERO1 PARAM PARAM PARAM PBAR +PB1F +PB2F PBAR +PB1W +PB2W RBAR RBAR RBAR RBAR RBAR RBAR RBAR RBAR RBAR RBAR SET1 SET1 SET1 SET1 SPC1 SPLINE2 +SPRC SPLINE2 +SPRW SPLINE2 +SPLC SPLINE2 +SPLW SPLINE2 +SP2FI SUPORT TRIM +TR1A +TR1B TRIM +TR2A +TR2B TRIM +TR3A +TR3B TRIM +TR4A +TR4B TRIM +TR5A +TR5B ENDDATA TOTAL COUNT=

122 210 211 212 220 221 222 310 311 312 1 2 4 1000 AUNITS GRDPNT WTMASS 100 1.0

1.44+9 1.44+9 110

23.83975+15. 27.11325-5. 24.61325-5. 29.61325-5. 21.33975-15. 18.83975-15. 23.83975-15. 32.88675+0. 30.38675+0. 35.38675+0. 5.40+8 5.40+8 120 210

.031081 90 .031081 1 4.0 1.0 1.0 0.0 101 2 1.5 0.5 3.0 0.5 0.0 111 110 111 112 110 112 121 120 121 122 120 122 211 210 211 212 210 212 221 220 221 222 220 222 311 310 311 312 310 312 1000 98 99 1100 99 100 2100 99 100 3100 99 100 1 1 90 1501 1000 1000 1. -1. 1601 1100 1100 -1. -1. 2501 2000 2000 1. -1. 2601 2100 2100 -1. -1. 3100 3100 3100 -1. -1. 90 23456 1 0.9 1200.0 URDD5 0. AILERON URDD4 0. URDD6 2 0.9 1200.0 URDD5 0. AILERON URDD4 0. URDD6 3 0.9 1200.0 URDD5 0. AILERON URDD2 0. URDD6 4 0.9 1200.0 PITCH 0. SIDES AILERON 0. RUDDER 5 0.9 1200.0 URDD5 0. YAW URDD4 0. URDD6

0. 0. 0. 0. 0. 0. 0. 5. 5. 5. 0.0 220

310

.347222 .30 -1.0 -1.0

1.0 1.0

-1.0

-1.0

+PB1F +PB2F

0.173611+2.0 -3.0 -0.5

0.462963 3.0 -0.5

-3.0

+PB1W +PB2W

123456 123456 123456 123456 123456 123456 123456 123456 123456 123456 111 211 311

112 212 312

121 221

122 222

1007

1000

0.

1.

1

+SPRC

1131

1100

0.

1.

2

+SPRW

2007

1000

0.

1.

1

+SPLC

2131

2100

0.

1.

20

+SPLW

3115

3100

0.

1.

300

+SP2FI

PITCH 0. 0. PITCH .436332 0. PITCH .436332 0. ANGLEA 0. .436332 PITCH .5435-3 0.

0. RUDDER

URDD3 0.

-1.0 URDD2

0.

+TR1A +TR1B

6.0499-4URDD3 YAW 0.

-4.8 URDD2

0.

+TR2A +TR2B

6.0499-4URDD3 ROLL 0.

-4.8 YAW

0.

+TR3A +TR3B

.0031512ELEV ROLL 0.

.436332 YAW 0.

+TR4A +TR4B

.2222-3 URDD3 -1.9170 ROLL -.3222-4URDD2 0.

+TR5A +TR5B

143

Listing 6-21. Sorted Bulk Data Entries for FSW Airplane in Unsymmetric Maneuvers EXAMPLE HA144E: 30 DEG FWD SWEPT WING WITH 3 CONTROLS PAGE 19 UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 90 M O * 1.610000E+04 0.000000E+00 0.000000E+00 0.000000E+00 5.000000E+02 0.000000E+00 * 0.000000E+00 1.610000E+04 0.000000E+00 -5.000000E+02 0.000000E+00 3.664468E+04 * 0.000000E+00 0.000000E+00 1.610000E+04 0.000000E+00 -3.664468E+04 0.000000E+00 * 0.000000E+00 -5.000000E+02 0.000000E+00 5.025000E+05 -2.743320E-12 -8.693375E+03 * 5.000000E+02 0.000000E+00 -3.664468E+04 -2.743320E-12 1.895271E+06 0.000000E+00 * 0.000000E+00 3.664468E+04 0.000000E+00 -8.693375E+03 0.000000E+00 2.392771E+06 S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 1.610000E+04 0.000000E+00 0.000000E+00 3.105590E-02 Y 1.610000E+04 2.276067E+00 0.000000E+00 3.105590E-02 Z 1.610000E+04 2.276067E+00 0.000000E+00 0.000000E+00 I(S) * 5.024845E+05 2.743320E-12 7.555341E+03 * * 2.743320E-12 1.811849E+06 0.000000E+00 * 7.555341E+03 0.000000E+00 2.309365E+06 * NX Nastran Aeroelastic Analysis User’s*Guide 6-88 I(Q) * 2.309396E+06 *

* * * * * *

Static Aeroelastic Analysis Sample Problems

* *

1.811849E+06

* 5.024528E+05 *

* 4.181319E-03 * 0.000000E+00 * -9.999912E-01

Q 0.000000E+00 1.000000E+00 0.000000E+00

9.999912E-01 * 0.000000E+00 * 4.181319E-03 *

FULL-SPAN MODEL N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T D E R I V A T I V E C O E F F I C I E N T S MACH = 9.0000E-01 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS TRIM VARIABLE COEFFICIENT RIGID UNSPLINED SPLINED INTERCEPT CX 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 ANGLEA CX 0.000000E+00 0.000000E+00 CY -2.234378E-09 -2.234378E-09 CZ -5.070976E+00 -5.070976E+00 CMX 1.283833E-08 1.283833E-08 CMY -2.870931E+00 -2.870931E+00 CMZ 1.899695E-09 1.899696E-09 PITCH CX 0.000000E+00 0.000000E+00 CY -2.518316E-08 -2.518316E-08 CZ -1.207429E+01 -1.207429E+01 CMX 5.162128E-08 5.162128E-08 CMY -9.953999E+00 -9.953999E+00 CMZ 1.785854E-08 1.785855E-08 URDD3 CX 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 URDD5 CX 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 CMX 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 ELEV

SIDES

YAW

ROLL

URDD2

URDD4

URDD6

AILERON

CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ CX CY

0.000000E+00 -1.429685E-08 -2.461396E-01 1.660761E-09 5.715299E-01 3.765754E-09 0.000000E+00 -7.158447E-01 -2.789167E-08 -3.276105E-02 -3.088518E-08 2.592296E-01 0.000000E+00 7.233017E-01 2.641076E-08 4.298553E-02 3.223686E-08 -2.775080E-01 0.000000E+00 7.965026E-02 -5.020057E-08 -4.184672E-01 3.861513E-08 -2.605253E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.082397E-01

0.000000E+00 -1.429685E-08 -2.461396E-01 1.660761E-09 5.715300E-01 3.765754E-09 0.000000E+00 -7.158447E-01 -2.789167E-08 -3.276105E-02 -2.386878E-08 2.592296E-01 0.000000E+00 7.233017E-01 2.641076E-08 4.298553E-02 2.676262E-08 -2.775080E-01 0.000000E+00 7.965026E-02 -5.020057E-08 -4.184672E-01 1.440686E-08 -2.605253E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.082397E-01

SUBCASE 1 R O L Q = 1.2000E+03 { 1.5000E+01 } { 0.0000E+00 } { 0.0000E+00 } ELASTIC RESTRAINED UNRESTRAINED 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.806870E-09 8.062219E-09 -6.462928E+00 -7.788311E+00 1.091487E-08 1.202968E-08 -3.667061E+00 -4.589579E+00 -5.304986E-10 -1.447366E-09 0.000000E+00 0.000000E+00 -1.639074E-08 -7.871786E-09 -1.285577E+01 -1.616588E+01 5.135293E-08 5.086361E-08 -1.027400E+01 -1.254529E+01 1.372134E-08 1.210880E-08 0.000000E+00 0.000000E+00 -9.429442E-12 0.000000E+00 3.809273E-03 0.000000E+00 -8.711054E-12 0.000000E+00 2.737913E-03 0.000000E+00 1.318927E-12 0.000000E+00 0.000000E+00 0.000000E+00 -1.529144E-10 0.000000E+00 7.602825E-02 0.000000E+00 -1.348617E-10 0.000000E+00 4.935852E-02 0.000000E+00 3.228951E-11 0.000000E+00 0.000000E+00 -1.293225E-08 -5.429845E-01 1.329735E-09 3.860484E-01 3.320143E-09 0.000000E+00 -6.697496E-01 -2.689029E-08 -2.824561E-02 -2.273260E-08 2.425404E-01 0.000000E+00 6.675829E-01 2.099408E-08 3.858262E-02 2.267642E-08 -2.572732E-01 0.000000E+00 1.209624E-01 -1.362767E-07 -5.070223E-01 -3.228925E-08 -4.053827E-02 0.000000E+00 7.188179E-04 2.894271E-11 3.018458E-05 2.437238E-11 -2.601771E-04 0.000000E+00 2.021708E-03 -1.250726E-09 -3.636955E-03 -5.123050E-10 -7.117986E-04 0.000000E+00 -2.213319E-02 -8.478067E-10 -9.981033E-04 -7.639817E-10 8.080994E-03 0.000000E+00 -1.142208E-01

0.000000E+00 -1.357400E-08 -5.180751E-01 1.130246E-09 3.979848E-01 3.563355E-09 0.000000E+00 -7.259906E-01 -3.533381E-08 -2.706761E-02 -2.927150E-08 2.630059E-01 0.000000E+00 7.284539E-01 2.885544E-08 3.630287E-02 2.890590E-08 -2.794171E-01 0.000000E+00 9.466008E-02 -1.260261E-07 -4.448231E-01 -3.193764E-08 -3.137653E-02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.026434E-01

NX Nastran Aeroelastic Analysis User’s Guide

6-89

Static Aeroelastic Analysis Sample Problems

Chapter 6

CZ CMX CMY CMZ CX CY CZ CMX CMY CMZ

RUDDER

1.223314E-07 2.747558E-01 6.498798E-08 3.948339E-02 0.000000E+00 3.491402E-01 9.397452E-09 3.745068E-02 1.250881E-08 -1.706690E-01

INTERMEDIATE MATRIX ... HP COLUMN 1 4.021594E-10 1.407690E+00 -9.998472E-09

2.974801E+00

-8.257486E-13

5.589295E-03

-2.190958E-11

-7.228872E-04

-2.208870E-09

5.403202E-02

-1.678953E-01

6.462641E-09

1.784256E-01

-5.601964E-09

3.155028E-02

2.087690E-08

-5.583723E-03

-6.944514E-12

3.735903E-04

2.159876E-10

1.906394E-02

2.094061E-10

-3.053800E-02

-4.163063E-08

1.061561E-01

-1.208490E-09

A E R O S T A T I C

1.223314E-07 2.747558E-01 6.015910E-08 3.948339E-02 0.000000E+00 3.491402E-01 9.397452E-09 3.745068E-02 1.212209E-08 -1.706690E-01

1.907469E-07 2.993375E-01 9.951416E-08 4.149461E-02 0.000000E+00 2.996780E-01 3.419316E-09 3.507515E-02 7.706671E-09 -1.525649E-01

-5.855442E-09

-7.312004E-02 COLUMN 2 -3.697710E-08 -2.120767E-01 COLUMN 3 5.549625E-12 -1.224622E-04 COLUMN 4 8.050897E-11 -7.453961E-03 COLUMN 5 -5.456374E-11 6.722745E-03 COLUMN 6 6.878061E-03 -4.676101E-10 COLUMN 7 -1.403152E-02 4.994862E-10 COLUMN 8 4.272658E-01 -1.371366E-10 COLUMN 9 1.048801E-05 5.006829E-13 COLUMN 10 2.427399E-02 -7.083333E-12 COLUMN 11 -2.522435E-05 -1.604130E-11 COLUMN 12 -2.377971E-01 1.762600E-09 COLUMN 13 -1.858679E-02 1.890525E-10

2.114289E-07 2.624515E-01 1.176604E-07 3.752521E-02 0.000000E+00 3.381170E-01 5.637137E-09 3.228774E-02 9.856436E-09 -1.665399E-01

-2.556327E-11 -2.831717E-09 -3.008684E-15 -2.549860E-12 -3.931490E-10 -3.704068E-02 4.077220E-02 6.442385E-03 -3.089367E-04 5.812574E-05 1.645562E-02 -6.602197E-03 2.767085E-02

R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 3.151205E-03 PITCH 0.000000E+00 URDD3 -1.000000E+00 URDD5 0.000000E+00 ELEV 1.724982E-02 SIDES -1.457720E-09 YAW -1.182399E-09 ROLL 1.214894E-10 URDD2 0.000000E+00 URDD4 0.000000E+00 URDD6 0.000000E+00 AILERON 0.000000E+00 RUDDER 0.000000E+00 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 1.480992E-01 1.777191E+02 1000 LS 1.110744E+03 6.942151E+02 2 LS 2.723155E-02 3.267786E+01 1001 LS 2.042366E+02 1.276479E+02 3 LS 1.154897E-02 1.385876E+01 1002 LS 8.661728E+01 5.413580E+01 4 LS 9.633810E-03 1.156057E+01 1003 LS 7.225358E+01 4.515849E+01 5 LS 1.172538E-01 1.407045E+02 1004 LS 8.794033E+02 5.496271E+02 6 LS 1.959021E-02 2.350826E+01 1005 LS 1.469266E+02 9.182912E+01 7 LS 8.839857E-03 1.060783E+01 1006 LS 6.629893E+01 4.143683E+01 8 LS 8.520475E-03 1.022457E+01 1007 LS 6.390356E+01 3.993972E+01 9 LS -3.474722E-02 -4.169667E+01 1100 LS -2.606042E+02 -1.628776E+02 10 LS -3.303323E-03 -3.963988E+00 1101 LS -2.477492E+01 -1.548433E+01 11 LS -6.613627E-04 -7.936352E-01 1102 LS -4.960220E+00 -3.100138E+00 12 LS -1.289034E-04 -1.546841E-01 1103 LS -9.667754E-01 -6.042346E-01 13 LS 7.057209E-03 8.468651E+00 1104 LS 5.292907E+01 3.308067E+01 14 LS -2.407379E-03 -2.888855E+00 1105 LS -1.805535E+01 -1.128459E+01 15 LS -2.951239E-04 -3.541487E-01 1106 LS -2.213429E+00 -1.383394E+00 16 LS -1.041256E-03 -1.249507E+00 1107 LS -7.809415E+00 -4.880884E+00 17 LS 1.412354E-01 1.694825E+02 1108 LS 1.059265E+03 6.620409E+02 18 LS 1.587099E-02 1.904519E+01 1109 LS 1.190325E+02 7.439529E+01 19 LS 2.484603E-03 2.981524E+00 1110 LS 1.863452E+01 1.164658E+01 20 LS -2.250555E-03 -2.700666E+00 1111 LS -1.687916E+01 -1.054948E+01 21 LS 1.082991E-01 1.299589E+02 1112 LS 8.122433E+02 5.076521E+02 22 LS 3.481510E-02 4.177812E+01 1113 LS 2.611133E+02 1.631958E+02 23 LS 9.695046E-03 1.163406E+01 1114 LS 7.271284E+01 4.544553E+01 24 LS 2.207176E-03 2.648612E+00 1115 LS 1.655382E+01 1.034614E+01 25 LS 8.606955E-02 1.032835E+02 1116 LS 6.455216E+02 4.034510E+02 3.688312E-02 1.592299E-02 5.199344E-03 6.924295E-02 3.236257E-02

D A T A

26 27 28 29 30

LS LS LS LS LS

4.425974E+01 1.910759E+01 6.239213E+00 8.309155E+01 3.883508E+01

6-90

NX Nastran Aeroelastic Analysis User’s Guide

1117 1118 1119 1120 1121

LS LS LS LS LS

2.766234E+02 1.194224E+02 3.899508E+01 5.193221E+02 2.427193E+02

1.728896E+02 7.463902E+01 2.437193E+01 3.245764E+02 1.516995E+02

Static Aeroelastic Analysis Sample Problems

31 LS 1.798351E-02 2.158021E+01 1122 LS 1.348763E+02 8.429770E+01 32 LS 7.599713E-03 9.119656E+00 1123 LS 5.699784E+01 3.562365E+01 33 LS 5.357746E-02 6.429295E+01 1124 LS 4.018310E+02 2.511443E+02 34 LS 2.556036E-02 3.067244E+01 1125 LS 1.917027E+02 1.198142E+02 35 LS 1.627770E-02 1.953324E+01 1126 LS 1.220827E+02 7.630171E+01 36 LS 8.524692E-03 1.022963E+01 1127 LS 6.393519E+01 3.995949E+01 37 LS 3.548426E-02 4.258111E+01 1128 LS 2.661319E+02 1.663325E+02 38 LS 1.658309E-02 1.989970E+01 1129 LS 1.243731E+02 7.773322E+01 39 LS 1.138929E-02 1.366715E+01 1130 LS 8.541969E+01 5.338730E+01 40 LS 7.125350E-03 8.550420E+00 1131 LS 5.344012E+01 3.340008E+01 41 LS 1.172538E-01 1.407045E+02 2000 LS 8.794033E+02 5.496271E+02 42 LS 1.959022E-02 2.350826E+01 2001 LS 1.469266E+02 9.182914E+01 43 LS 8.839858E-03 1.060783E+01 2002 LS 6.629893E+01 4.143683E+01 44 LS 8.520480E-03 1.022458E+01 2003 LS 6.390360E+01 3.993975E+01 45 LS 1.480992E-01 1.777191E+02 2004 LS 1.110744E+03 6.942150E+02 46 LS 2.723155E-02 3.267786E+01 2005 LS 2.042366E+02 1.276479E+02 47 LS 1.154897E-02 1.385876E+01 2006 LS 8.661728E+01 5.413580E+01 48 LS 9.633814E-03 1.156058E+01 2007 LS 7.225360E+01 4.515850E+01 49 LS 3.548426E-02 4.258112E+01 2100 LS 2.661320E+02 1.663325E+02 50 LS 1.658307E-02 1.989969E+01 2101 LS 1.243731E+02 7.773318E+01 51 LS 1.138929E-02 1.366715E+01 2102 LS 8.541971E+01 5.338732E+01 52 LS 7.125352E-03 8.550422E+00 2103 LS 5.344014E+01 3.340009E+01 53 LS 5.357747E-02 6.429296E+01 2104 LS 4.018310E+02 2.511444E+02 54 LS 2.556036E-02 3.067244E+01 2105 LS 1.917027E+02 1.198142E+02 55 LS 1.627768E-02 1.953322E+01 2106 LS 1.220826E+02 7.630165E+01 56 LS 8.524698E-03 1.022964E+01 2107 LS 6.393523E+01 3.995952E+01 57 LS 6.924295E-02 8.309154E+01 2108 LS 5.193221E+02 3.245763E+02 58 LS 3.236258E-02 3.883509E+01 2109 LS 2.427194E+02 1.516996E+02 59 LS 1.798349E-02 2.158019E+01 2110 LS 1.348762E+02 8.429765E+01 60 LS 7.599720E-03 9.119664E+00 2111 LS 5.699791E+01 3.562369E+01 61 LS 8.606955E-02 1.032835E+02 2112 LS 6.455215E+02 4.034510E+02 62 LS 3.688312E-02 4.425974E+01 2113 LS 2.766234E+02 1.728896E+02 63 LS 1.592300E-02 1.910760E+01 2114 LS 1.194225E+02 7.463905E+01 64 LS 5.199342E-03 6.239211E+00 2115 LS 3.899507E+01 2.437192E+01 65 LS 1.082991E-01 1.299590E+02 2116 LS 8.122435E+02 5.076522E+02 66 LS 3.481508E-02 4.177809E+01 2117 LS 2.611131E+02 1.631957E+02 67 LS 9.695059E-03 1.163407E+01 2118 LS 7.271294E+01 4.544559E+01 68 LS 2.207176E-03 2.648611E+00 2119 LS 1.655382E+01 1.034614E+01 69 LS 1.412354E-01 1.694825E+02 2120 LS 1.059266E+03 6.620410E+02 70 LS 1.587098E-02 1.904517E+01 2121 LS 1.190323E+02 7.439520E+01 71 LS 2.484598E-03 2.981517E+00 2122 LS 1.863449E+01 1.164656E+01 72 LS -2.250552E-03 -2.700662E+00 2123 LS -1.687914E+01 -1.054947E+01 73 LS 7.057216E-03 8.468659E+00 2124 LS 5.292912E+01 3.308070E+01 74 LS -2.407386E-03 -2.888864E+00 2125 LS -1.805540E+01 -1.128462E+01 75 LS -2.951450E-04 -3.541740E-01 2126 LS -2.213588E+00 -1.383493E+00 76 LS -1.041241E-03 -1.249489E+00 2127 LS -7.809305E+00 -4.880816E+00 77 LS -3.474717E-02 -4.169661E+01 2128 LS -2.606038E+02 -1.628774E+02 78 LS -3.303373E-03 -3.964047E+00 2129 LS -2.477530E+01 -1.548456E+01 79 LS -6.613678E-04 -7.936413E-01 2130 LS -4.960258E+00 -3.100161E+00 80 LS -1.289032E-04 -1.546838E-01 2131 LS -9.667740E-01 -6.042337E-01 81 LS 1.526758E-09 1.832109E-06 3100 LS 1.145068E-05 7.156677E-06 82 LS -1.036260E-09 -1.243512E-06 3101 LS -7.771948E-06 -4.857468E-06 83 LS -1.645607E-11 -1.974728E-08 3102 LS -1.234205E-07 -7.713783E-08 84 LS -5.575136E-10 -6.690163E-07 3103 LS -4.181352E-06 -2.613346E-06 85 LS 3.053273E-09 3.663927E-06 3104 LS 2.289954E-05 1.431222E-05 86 LS -2.462315E-09 -2.954778E-06 3105 LS -1.846737E-05 -1.154210E-05 87 LS -2.511690E-10 -3.014028E-07 3106 LS -1.883768E-06 -1.177355E-06 88 LS -8.563940E-10 -1.027673E-06 3107 LS -6.422955E-06 -4.014347E-06 89 LS 1.070075E-09 1.284090E-06 3108 LS 8.025565E-06 5.015978E-06 90 LS 3.735966E-10 4.483159E-07 3109 LS 2.801974E-06 1.751234E-06 91 LS -5.124166E-10 -6.148999E-07 3110 LS -3.843124E-06 -2.401953E-06 92 LS -4.349331E-10 -5.219197E-07 3111 LS -3.261998E-06 -2.038749E-06 93 LS -1.056676E-08 -1.268011E-05 3112 LS -7.925069E-05 -4.953168E-05 94 LS 1.122110E-08 1.346532E-05 3113 LS 8.415823E-05 5.259889E-05 95 LS -1.116792E-09 -1.340150E-06 3114 LS -8.375938E-06 -5.234961E-06 96 LS 5.662081E-10 6.794498E-07 3115 LS 4.246561E-06 2.654101E-06 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. FULL-SPAN MODEL SUBCASE 2 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 1.370419E-02 PITCH 6.049900E-04 URDD3 -4.800000E+00 URDD5 0.000000E+00 ELEV 8.539610E-02 SIDES -1.823546E-02 YAW 0.000000E+00 ROLL 2.600431E-01 URDD2 0.000000E+00 URDD4 0.000000E+00 URDD6 -2.168404E-19 AILERON 4.363320E-01 RUDDER 2.058725E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2)

NX Nastran Aeroelastic Analysis User’s Guide

6-91

Chapter 6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

6-92

LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS

Static Aeroelastic Analysis Sample Problems

7.864448E-01 1.396064E-01 6.360418E-02 5.981490E-02 7.315380E-01 1.086405E-01 5.830283E-02 7.348040E-02 -4.793925E-02 1.212338E-02 -2.603003E-02 -3.782949E-03 3.487090E-01 -4.106984E-02 -6.558547E-02 -4.328452E-02 1.517685E+00 7.238707E-02 -1.522643E-01 -1.218619E-01 1.587969E+00 2.385288E-01 -2.716918E-01 -3.019391E-01 1.673347E+00 3.382304E-01 -3.085978E-01 -1.427823E+00 1.705563E+00 4.176543E-01 -2.567720E-01 -1.599504E+00 1.602263E+00 4.607435E-01 -1.318212E-01 -1.564726E+00 1.246066E+00 3.864193E-01 4.524189E-02 -1.317840E+00 4.035736E-01 8.393354E-02 3.022441E-02 1.113158E-02 6.473656E-01 1.277147E-01 5.201303E-02 3.632584E-02 -9.168379E-01 -2.280832E-01 6.587906E-02 1.388865E+00 -1.101580E+00 -2.160054E-01 2.907227E-01 1.649767E+00 -1.054028E+00 -1.068757E-01 4.325479E-01 1.675814E+00 -8.580897E-01 1.679588E-02 4.646642E-01 1.481088E+00 -5.555010E-01 9.684183E-02 3.676826E-01 3.265942E-01 -1.633413E-01 7.936113E-02 1.791099E-01 1.042501E-01 -2.986870E-01 1.669648E-02 6.635843E-02 3.727235E-02 -3.086562E-01 -4.308398E-02 2.404013E-02 6.305821E-03 -2.999334E-01 -2.692820E-02 9.641200E-03 6.186832E-02 -1.522581E-01 -6.095775E-02 1.336691E-03 7.109066E-02 8.106790E-02 -5.076220E-02 -1.349962E-02

9.437338E+02 1.675277E+02 7.632501E+01 7.177788E+01 8.778456E+02 1.303686E+02 6.996339E+01 8.817648E+01 -5.752710E+01 1.454806E+01 -3.123603E+01 -4.539539E+00 4.184508E+02 -4.928381E+01 -7.870257E+01 -5.194143E+01 1.821222E+03 8.686449E+01 -1.827172E+02 -1.462342E+02 1.905563E+03 2.862346E+02 -3.260302E+02 -3.623269E+02 2.008017E+03 4.058765E+02 -3.703174E+02 -1.713388E+03 2.046675E+03 5.011851E+02 -3.081264E+02 -1.919405E+03 1.922716E+03 5.528923E+02 -1.581854E+02 -1.877671E+03 1.495279E+03 4.637032E+02 5.429027E+01 -1.581409E+03 4.842884E+02 1.007202E+02 3.626929E+01 1.335789E+01 7.768387E+02 1.532577E+02 6.241564E+01 4.359101E+01 -1.100205E+03 -2.736999E+02 7.905488E+01 1.666639E+03 -1.321896E+03 -2.592064E+02 3.488673E+02 1.979721E+03 -1.264833E+03 -1.282508E+02 5.190575E+02 2.010976E+03 -1.029708E+03 2.015505E+01 5.575970E+02 1.777306E+03 -6.666012E+02 1.162102E+02 4.412191E+02 3.919131E+02 -1.960096E+02 9.523335E+01 2.149318E+02 1.251001E+02 -3.584244E+02 2.003577E+01 7.963012E+01 4.472682E+01 -3.703875E+02 -5.170077E+01 2.884816E+01 7.566985E+00 -3.599201E+02 -3.231384E+01 1.156944E+01 7.424199E+01 -1.827097E+02 -7.314930E+01 1.604029E+00 8.530879E+01 9.728148E+01 -6.091463E+01 -1.619954E+01

NX Nastran Aeroelastic Analysis User’s Guide

1000 1001 1002 1003 1004 1005 1006 1007 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 2000 2001 2002 2003 2004 2005 2006 2007 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110

LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS

5.898336E+03 1.047048E+03 4.770313E+02 4.486117E+02 5.486535E+03 8.148035E+02 4.372712E+02 5.511030E+02 -3.595444E+02 9.092536E+01 -1.952252E+02 -2.837212E+01 2.615318E+03 -3.080238E+02 -4.918911E+02 -3.246339E+02 1.138264E+04 5.429031E+02 -1.141982E+03 -9.139639E+02 1.190977E+04 1.788966E+03 -2.037689E+03 -2.264543E+03 1.255010E+04 2.536728E+03 -2.314483E+03 -1.070867E+04 1.279172E+04 3.132407E+03 -1.925790E+03 -1.199628E+04 1.201698E+04 3.455577E+03 -9.886589E+02 -1.173545E+04 9.345496E+03 2.898145E+03 3.393142E+02 -9.883804E+03 3.026802E+03 6.295015E+02 2.266831E+02 8.348683E+01 4.855242E+03 9.578604E+02 3.900977E+02 2.724438E+02 -6.876284E+03 -1.710624E+03 4.940930E+02 1.041649E+04 -8.261851E+03 -1.620040E+03 2.180421E+03 1.237325E+04 -7.905208E+03 -8.015675E+02 3.244110E+03 1.256860E+04 -6.435673E+03 1.259691E+02 3.484981E+03 1.110816E+04 -4.166257E+03 7.263138E+02 2.757620E+03 2.449457E+03 -1.225060E+03 5.952085E+02 1.343324E+03 7.818759E+02 -2.240152E+03 1.252236E+02 4.976884E+02 2.795426E+02 -2.314922E+03 -3.231298E+02 1.803010E+02 4.729366E+01 -2.249500E+03 -2.019615E+02 7.230900E+01 4.640125E+02 -1.141936E+03 -4.571831E+02 1.002518E+01 5.331800E+02 6.080092E+02 -3.807165E+02 -1.012471E+02

3.686460E+03 6.544051E+02 2.981446E+02 2.803823E+02 3.429084E+03 5.092522E+02 2.732945E+02 3.444394E+02 -2.247152E+02 5.682835E+01 -1.220158E+02 -1.773257E+01 1.634574E+03 -1.925149E+02 -3.074320E+02 -2.028962E+02 7.114149E+03 3.393144E+02 -7.137390E+02 -5.712275E+02 7.443606E+03 1.118104E+03 -1.273556E+03 -1.415339E+03 7.843815E+03 1.585455E+03 -1.446552E+03 -6.692922E+03 7.994825E+03 1.957754E+03 -1.203619E+03 -7.497675E+03 7.510610E+03 2.159736E+03 -6.179119E+02 -7.334654E+03 5.840936E+03 1.811341E+03 2.120714E+02 -6.177377E+03 1.891751E+03 3.934384E+02 1.416769E+02 5.217927E+01 3.034526E+03 5.986628E+02 2.438111E+02 1.702774E+02 -4.297677E+03 -1.069140E+03 3.088081E+02 6.510307E+03 -5.163657E+03 -1.012525E+03 1.362763E+03 7.733284E+03 -4.940755E+03 -5.009798E+02 2.027569E+03 7.855376E+03 -4.022296E+03 7.873070E+01 2.178113E+03 6.942601E+03 -2.603911E+03 4.539461E+02 1.723512E+03 1.530911E+03 -7.656625E+02 3.720053E+02 8.395778E+02 4.886725E+02 -1.400095E+03 7.826473E+01 3.110553E+02 1.747141E+02 -1.446826E+03 -2.019561E+02 1.126881E+02 2.955853E+01 -1.405938E+03 -1.262259E+02 4.519312E+01 2.900078E+02 -7.137099E+02 -2.857395E+02 6.265737E+00 3.332375E+02 3.800058E+02 -2.379478E+02 -6.327945E+01

Static Aeroelastic Analysis Sample Problems

92 LS 6.774797E-02 8.129756E+01 3111 LS 5.081098E+02 3.175686E+02 93 LS 2.938147E-01 3.525776E+02 3112 LS 2.203610E+03 1.377256E+03 94 LS -8.344214E-03 -1.001306E+01 3113 LS -6.258160E+01 -3.911350E+01 95 LS -2.760044E-02 -3.312052E+01 3114 LS -2.070033E+02 -1.293770E+02 96 LS 5.371650E-02 6.445979E+01 3115 LS 4.028737E+02 2.517961E+02 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. FULL-SPAN MODEL SUBCASE 3 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 1.370419E-02 PITCH 6.049900E-04 URDD3 -4.800000E+00 URDD5 0.000000E+00 ELEV 8.539611E-02 SIDES -6.297442E-02 YAW 9.658545E-19 ROLL 0.000000E+00 URDD2 0.000000E+00 URDD4 4.454218E+00 URDD6 -2.168404E-19 AILERON 4.363320E-01 RUDDER 1.099768E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 7.164526E-01 8.597431E+02 1000 LS 5.373395E+03 3.358371E+03 2 LS 1.330722E-01 1.596866E+02 1001 LS 9.980414E+02 6.237759E+02 3 LS 5.625242E-02 6.750291E+01 1002 LS 4.218932E+02 2.636832E+02 4 LS 4.402409E-02 5.282891E+01 1003 LS 3.301807E+02 2.063629E+02 5 LS 5.665109E-01 6.798130E+02 1004 LS 4.248832E+03 2.655520E+03 6 LS 9.494174E-02 1.139301E+02 1005 LS 7.120630E+02 4.450394E+02 7 LS 4.063968E-02 4.876761E+01 1006 LS 3.047976E+02 1.904985E+02 8 LS 3.221849E-02 3.866218E+01 1007 LS 2.416386E+02 1.510242E+02 9 LS -2.400071E-01 -2.880085E+02 1100 LS -1.800053E+03 -1.125033E+03 10 LS -5.004971E-02 -6.005965E+01 1101 LS -3.753728E+02 -2.346080E+02 11 LS -3.032419E-02 -3.638903E+01 1102 LS -2.274315E+02 -1.421447E+02 12 LS -3.890117E-03 -4.668140E+00 1103 LS -2.917587E+01 -1.823492E+01 13 LS -1.212328E-01 -1.454794E+02 1104 LS -9.092462E+02 -5.682789E+02 14 LS -1.437190E-01 -1.724628E+02 1105 LS -1.077892E+03 -6.736827E+02 15 LS -9.253818E-02 -1.110458E+02 1106 LS -6.940364E+02 -4.337728E+02 16 LS -4.014689E-02 -4.817627E+01 1107 LS -3.011017E+02 -1.881885E+02 17 LS 4.887081E-01 5.864497E+02 1108 LS 3.665311E+03 2.290819E+03 18 LS -1.470592E-01 -1.764710E+02 1109 LS -1.102944E+03 -6.893400E+02 19 LS -2.159077E-01 -2.590893E+02 1110 LS -1.619308E+03 -1.012068E+03 20 LS -1.244790E-01 -1.493748E+02 1111 LS -9.335923E+02 -5.834952E+02 21 LS 3.513409E-01 4.216090E+02 1112 LS 2.635056E+03 1.646910E+03 22 LS -1.128529E-01 -1.354234E+02 1113 LS -8.463965E+02 -5.289978E+02 23 LS -3.923257E-01 -4.707908E+02 1114 LS -2.942443E+03 -1.839027E+03 24 LS -3.349355E-01 -4.019226E+02 1115 LS -2.512016E+03 -1.570010E+03 25 LS 2.979548E-01 3.575457E+02 1116 LS 2.234661E+03 1.396663E+03 26 LS -1.002865E-01 -1.203438E+02 1117 LS -7.521486E+02 -4.700929E+02 27 LS -4.860806E-01 -5.832968E+02 1118 LS -3.645605E+03 -2.278503E+03 28 LS -1.488213E+00 -1.785856E+03 1119 LS -1.116160E+04 -6.975999E+03 29 LS 2.788988E-01 3.346785E+02 1120 LS 2.091741E+03 1.307338E+03 30 LS -6.212339E-02 -7.454807E+01 1121 LS -4.659254E+02 -2.912034E+02 31 LS -4.741541E-01 -5.689850E+02 1122 LS -3.556156E+03 -2.222598E+03 32 LS -1.685115E+00 -2.022138E+03 1123 LS -1.263836E+04 -7.898977E+03 33 LS 2.503476E-01 3.004172E+02 1124 LS 1.877607E+03 1.173505E+03 34 LS -2.328361E-03 -2.794033E+00 1125 LS -1.746271E+01 -1.091419E+01 35 LS -3.588144E-01 -4.305772E+02 1126 LS -2.691108E+03 -1.681942E+03 36 LS -1.666845E+00 -2.000214E+03 1127 LS -1.250134E+04 -7.813336E+03 37 LS 1.853577E-01 2.224293E+02 1128 LS 1.390183E+03 8.688643E+02 38 LS 3.228679E-02 3.874414E+01 1129 LS 2.421509E+02 1.513443E+02 39 LS -1.375535E-01 -1.650642E+02 1130 LS -1.031651E+03 -6.447819E+02 40 LS -1.411519E+00 -1.693823E+03 1131 LS -1.058639E+04 -6.616495E+03 41 LS 5.686010E-01 6.823212E+02 2000 LS 4.264507E+03 2.665317E+03 42 LS 9.763229E-02 1.171587E+02 2001 LS 7.322422E+02 4.576514E+02 43 LS 4.788753E-02 5.746504E+01 2002 LS 3.591565E+02 2.244728E+02 44 LS 5.239346E-02 6.287215E+01 2003 LS 3.929509E+02 2.455943E+02 45 LS 7.173581E-01 8.608297E+02 2004 LS 5.380186E+03 3.362616E+03 46 LS 1.342490E-01 1.610988E+02 2005 LS 1.006867E+03 6.292921E+02 47 LS 5.936474E-02 7.123769E+01 2006 LS 4.452356E+02 2.782722E+02 48 LS 5.211661E-02 6.253994E+01 2007 LS 3.908746E+02 2.442966E+02 49 LS 1.438708E-01 1.726450E+02 2100 LS 1.079031E+03 6.743944E+02 50 LS 1.260490E-01 1.512588E+02 2101 LS 9.453678E+02 5.908550E+02 51 LS 2.486745E-01 2.984094E+02 2102 LS 1.865059E+03 1.165662E+03 52 LS 1.482544E+00 1.779053E+03 2103 LS 1.111908E+04 6.949425E+03 53 LS 2.503359E-01 3.004030E+02 2104 LS 1.877519E+03 1.173449E+03 54 LS 2.470665E-01 2.964799E+02 2105 LS 1.852999E+03 1.158125E+03 55 LS 5.177157E-01 6.212589E+02 2106 LS 3.882868E+03 2.426793E+03 56 LS 1.751886E+00 2.102263E+03 2107 LS 1.313915E+04 8.211967E+03 57 LS 3.726362E-01 4.471634E+02 2108 LS 2.794771E+03 1.746732E+03 58 LS 3.729022E-01 4.474826E+02 2109 LS 2.796766E+03 1.747979E+03 59 LS 6.499299E-01 7.799159E+02 2110 LS 4.874475E+03 3.046547E+03 60 LS 1.761425E+00 2.113710E+03 2111 LS 1.321069E+04 8.256679E+03 61 LS 5.173029E-01 6.207635E+02 2112 LS 3.879772E+03 2.424858E+03

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LS LS LS LS LS LS LS LS LS LS LS LS LS LS

Static Aeroelastic Analysis Sample Problems

4.553127E-01 6.421471E-01 1.541478E+00 6.811275E-01 4.482233E-01 4.883165E-01 3.595906E-01 8.656359E-01 2.988071E-01 2.427534E-01 1.068672E-01 1.712547E-01 1.193454E-01 9.331115E-02

5.463752E+02 7.705765E+02 1.849773E+03 8.173531E+02 5.378679E+02 5.859798E+02 4.315088E+02 1.038763E+03 3.585685E+02 2.913040E+02 1.282406E+02 2.055057E+02 1.432145E+02 1.119734E+02

2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126

LS LS LS LS LS LS LS LS LS LS LS LS LS LS

3.414845E+03 4.816103E+03 1.156108E+04 5.108457E+03 3.361674E+03 3.662374E+03 2.696930E+03 6.492269E+03 2.241053E+03 1.820651E+03 8.015040E+02 1.284411E+03 8.950905E+02 6.998337E+02

2.134279E+03 3.010064E+03 7.225677E+03 3.192786E+03 2.101046E+03 2.288984E+03 1.685582E+03 4.057668E+03 1.400658E+03 1.137907E+03 5.009401E+02 8.027566E+02 5.594316E+02 4.373961E+02

76 LS 3.413485E-02 4.096181E+01 2127 LS 2.560114E+02 1.600071E+02 77 LS -1.165893E-01 -1.399072E+02 2128 LS -8.744197E+02 -5.465123E+02 78 LS 1.908966E-02 2.290759E+01 2129 LS 1.431724E+02 8.948277E+01 79 LS 2.833431E-02 3.400117E+01 2130 LS 2.125073E+02 1.328170E+02 80 LS 6.413082E-03 7.695698E+00 2131 LS 4.809811E+01 3.006132E+01 81 LS 7.503304E-02 9.003964E+01 3100 LS 5.627477E+02 3.517173E+02 82 LS 2.032789E-03 2.439347E+00 3101 LS 1.524592E+01 9.528701E+00 83 LS 7.339601E-03 8.807521E+00 3102 LS 5.504700E+01 3.440438E+01 84 LS 3.333901E-02 4.000681E+01 3103 LS 2.500426E+02 1.562766E+02 85 LS 5.730936E-02 6.877123E+01 3104 LS 4.298202E+02 2.686376E+02 86 LS 9.735225E-04 1.168227E+00 3105 LS 7.301420E+00 4.563387E+00 87 LS 6.672278E-03 8.006733E+00 3106 LS 5.004208E+01 3.127630E+01 88 LS 3.910184E-02 4.692221E+01 3107 LS 2.932639E+02 1.832899E+02 89 LS 4.101618E-02 4.921942E+01 3108 LS 3.076213E+02 1.922633E+02 90 LS -1.635940E-02 -1.963128E+01 3109 LS -1.226955E+02 -7.668471E+01 91 LS -1.008363E-03 -1.210036E+00 3110 LS -7.562724E+00 -4.726703E+00 92 LS 3.825928E-02 4.591113E+01 3111 LS 2.869446E+02 1.793404E+02 93 LS 4.080638E-02 4.896766E+01 3112 LS 3.060479E+02 1.912799E+02 94 LS -3.795005E-02 -4.554007E+01 3113 LS -2.846254E+02 -1.778909E+02 95 LS -1.964539E-02 -2.357447E+01 3114 LS -1.473405E+02 -9.208778E+01 96 LS 3.002778E-02 3.603334E+01 3115 LS 2.252084E+02 1.407552E+02 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. FULL-SPAN MODEL SUBCASE 4 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 3.151200E-03 PITCH 0.000000E+00 URDD3 -8.659549E+00 URDD5 5.768343E-01 ELEV 4.363320E-01 SIDES 0.000000E+00 YAW 1.135192E-19 ROLL -1.695235E-19 URDD2 3.127156E+00 URDD4 5.235149E-01 URDD6 -5.281315E-01 AILERON 0.000000E+00 RUDDER 4.363320E-01 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 3.127665E+00 3.753198E+03 1000 LS 2.345749E+04 1.466093E+04 2 LS 5.400828E-01 6.480993E+02 1001 LS 4.050621E+03 2.531638E+03 3 LS 1.769049E-01 2.122858E+02 1002 LS 1.326786E+03 8.292416E+02 4 LS 9.641417E-02 1.156970E+02 1003 LS 7.231063E+02 4.519414E+02 5 LS 2.476894E+00 2.972273E+03 1004 LS 1.857671E+04 1.161044E+04 6 LS 3.860235E-01 4.632282E+02 1005 LS 2.895176E+03 1.809485E+03 7 LS 1.326031E-01 1.591237E+02 1006 LS 9.945233E+02 6.215770E+02 8 LS 8.484723E-02 1.018167E+02 1007 LS 6.363542E+02 3.977214E+02 9 LS -2.170004E+00 -2.604005E+03 1100 LS -1.627503E+04 -1.017190E+04 10 LS -4.091441E-01 -4.909729E+02 1101 LS -3.068581E+03 -1.917863E+03 11 LS -5.153607E-02 -6.184329E+01 1102 LS -3.865206E+02 -2.415754E+02 12 LS 2.604646E-01 3.125575E+02 1103 LS 1.953484E+03 1.220927E+03 13 LS -1.106834E+00 -1.328201E+03 1104 LS -8.301254E+03 -5.188284E+03 14 LS -5.150145E-01 -6.180175E+02 1105 LS -3.862609E+03 -2.414131E+03 15 LS -1.648353E-01 -1.978024E+02 1106 LS -1.236265E+03 -7.726657E+02 16 LS 1.409162E-02 1.690994E+01 1107 LS 1.056871E+02 6.605445E+01 17 LS 1.700471E+00 2.040565E+03 1108 LS 1.275353E+04 7.970957E+03 18 LS -1.818285E-01 -2.181942E+02 1109 LS -1.363714E+03 -8.523212E+02 19 LS -1.830603E-01 -2.196723E+02 1110 LS -1.372952E+03 -8.580950E+02 20 LS -8.198647E-02 -9.838377E+01 1111 LS -6.148986E+02 -3.843116E+02 21 LS 1.153930E+00 1.384716E+03 1112 LS 8.654474E+03 5.409046E+03 22 LS 2.310237E-01 2.772285E+02 1113 LS 1.732678E+03 1.082924E+03 23 LS -7.476046E-02 -8.971255E+01 1114 LS -5.607034E+02 -3.504396E+02 24 LS -6.853464E-02 -8.224157E+01 1115 LS -5.140098E+02 -3.212561E+02 25 LS 8.178837E-01 9.814605E+02 1116 LS 6.134128E+03 3.833830E+03 26 LS 3.208675E-01 3.850410E+02 1117 LS 2.406507E+03 1.504067E+03 27 LS 5.670957E-02 6.805148E+01 1118 LS 4.253217E+02 2.658261E+02 28 LS -3.194197E-02 -3.833036E+01 1119 LS -2.395647E+02 -1.497280E+02 29 LS 6.001082E-01 7.201298E+02 1120 LS 4.500812E+03 2.813007E+03

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Static Aeroelastic Analysis Sample Problems

30 LS 2.867484E-01 3.440981E+02 1121 LS 2.150613E+03 1.344133E+03 31 LS 1.240202E-01 1.488243E+02 1122 LS 9.301517E+02 5.813448E+02 32 LS 1.601115E-02 1.921338E+01 1123 LS 1.200836E+02 7.505225E+01 33 LS 4.306050E-01 5.167260E+02 1124 LS 3.229537E+03 2.018461E+03 34 LS 2.209344E-01 2.651212E+02 1125 LS 1.657008E+03 1.035630E+03 35 LS 1.299042E-01 1.558850E+02 1126 LS 9.742812E+02 6.089257E+02 36 LS 4.789853E-02 5.747823E+01 1127 LS 3.592390E+02 2.245244E+02 37 LS 2.675137E-01 3.210165E+02 1128 LS 2.006353E+03 1.253971E+03 38 LS 1.388383E-01 1.666059E+02 1129 LS 1.041287E+03 6.508043E+02 39 LS 9.473918E-02 1.136870E+02 1130 LS 7.105438E+02 4.440899E+02 40 LS 5.058339E-02 6.070007E+01 1131 LS 3.793754E+02 2.371096E+02 41 LS 2.476673E+00 2.972008E+03 2000 LS 1.857505E+04 1.160941E+04 42 LS 3.857210E-01 4.628652E+02 2001 LS 2.892907E+03 1.808067E+03 43 LS 1.318253E-01 1.581903E+02 2002 LS 9.886895E+02 6.179309E+02 44 LS 8.277810E-02 9.933372E+01 2003 LS 6.208358E+02 3.880224E+02 45 LS 3.127570E+00 3.753083E+03 2004 LS 2.345677E+04 1.466048E+04 46 LS 5.399531E-01 6.479437E+02 2005 LS 4.049648E+03 2.531030E+03 47 LS 1.765791E-01 2.118949E+02 2006 LS 1.324343E+03 8.277144E+02 48 LS 9.560930E-02 1.147312E+02 2007 LS 7.170698E+02 4.481686E+02 49 LS 2.365708E-01 2.838850E+02 2100 LS 1.774281E+03 1.108926E+03 50 LS 1.260567E-01 1.512681E+02 2101 LS 9.454256E+02 5.908911E+02 51 LS 8.619065E-02 1.034288E+02 2102 LS 6.464299E+02 4.040187E+02 52 LS 4.422990E-02 5.307588E+01 2103 LS 3.317242E+02 2.073277E+02 53 LS 3.865022E-01 4.638026E+02 2104 LS 2.898766E+03 1.811729E+03 54 LS 2.022270E-01 2.426724E+02 2105 LS 1.516703E+03 9.479393E+02 55 LS 1.174357E-01 1.409229E+02 2106 LS 8.807682E+02 5.504802E+02 56 LS 3.880361E-02 4.656433E+01 2107 LS 2.910271E+02 1.818919E+02 57 LS 5.475923E-01 6.571108E+02 2108 LS 4.106942E+03 2.566839E+03 58 LS 2.645937E-01 3.175124E+02 2109 LS 1.984453E+03 1.240283E+03 59 LS 1.086402E-01 1.303683E+02 2110 LS 8.148018E+02 5.092512E+02 60 LS 3.197597E-03 3.837116E+00 2111 LS 2.398198E+01 1.498874E+01 61 LS 7.601801E-01 9.122161E+02 2112 LS 5.701351E+03 3.563344E+03 62 LS 2.962995E-01 3.555594E+02 2113 LS 2.222246E+03 1.388904E+03 63 LS 3.752650E-02 4.503180E+01 2114 LS 2.814487E+02 1.759054E+02 64 LS -5.221706E-02 -6.266048E+01 2115 LS -3.916280E+02 -2.447675E+02 65 LS 1.093825E+00 1.312590E+03 2116 LS 8.203690E+03 5.127306E+03 66 LS 2.045947E-01 2.455137E+02 2117 LS 1.534460E+03 9.590377E+02 67 LS -1.020614E-01 -1.224737E+02 2118 LS -7.654608E+02 -4.784130E+02 68 LS -1.066064E-01 -1.279277E+02 2119 LS -7.995480E+02 -4.997176E+02 69 LS 1.641414E+00 1.969697E+03 2120 LS 1.231061E+04 7.694129E+03 70 LS -2.116487E-01 -2.539784E+02 2121 LS -1.587365E+03 -9.921031E+02 71 LS -2.304658E-01 -2.765590E+02 2122 LS -1.728494E+03 -1.080309E+03 72 LS -1.621420E-01 -1.945704E+02 2123 LS -1.216065E+03 -7.600410E+02 73 LS -1.154681E+00 -1.385617E+03 2124 LS -8.660106E+03 -5.412566E+03 74 LS -5.556432E-01 -6.667719E+02 2125 LS -4.167324E+03 -2.604577E+03 75 LS -2.661954E-01 -3.194345E+02 2126 LS -1.996466E+03 -1.247792E+03 76 LS -1.939341E-01 -2.327210E+02 2127 LS -1.454506E+03 -9.090663E+02 77 LS -2.209841E+00 -2.651810E+03 2128 LS -1.657381E+04 -1.035863E+04 78 LS -4.982656E-01 -5.979187E+02 2129 LS -3.736992E+03 -2.335620E+03 79 LS -2.723013E-01 -3.267616E+02 2130 LS -2.042260E+03 -1.276412E+03 80 LS -3.711098E-01 -4.453317E+02 2131 LS -2.783323E+03 -1.739577E+03 81 LS 1.276761E-01 1.532113E+02 3100 LS 9.575705E+02 5.984816E+02 82 LS 2.003490E-01 2.404188E+02 3101 LS 1.502618E+03 9.391360E+02 83 LS 4.230284E-01 5.076341E+02 3102 LS 3.172713E+03 1.982946E+03 84 LS 1.376074E+00 1.651289E+03 3103 LS 1.032055E+04 6.450348E+03 85 LS 1.241964E-01 1.490357E+02 3104 LS 9.314728E+02 5.821705E+02 86 LS 2.328381E-01 2.794058E+02 3105 LS 1.746286E+03 1.091429E+03 87 LS 5.256876E-01 6.308252E+02 3106 LS 3.942657E+03 2.464161E+03 88 LS 1.658361E+00 1.990034E+03 3107 LS 1.243771E+04 7.773571E+03 89 LS 9.451665E-02 1.134200E+02 3108 LS 7.088749E+02 4.430468E+02 90 LS 2.021007E-01 2.425209E+02 3109 LS 1.515755E+03 9.473472E+02 91 LS 4.974560E-01 5.969472E+02 3110 LS 3.730920E+03 2.331825E+03 92 LS 1.704293E+00 2.045152E+03 3111 LS 1.278220E+04 7.988874E+03 93 LS 6.307691E-02 7.569229E+01 3112 LS 4.730768E+02 2.956730E+02 94 LS 1.498650E-01 1.798380E+02 3113 LS 1.123987E+03 7.024921E+02 95 LS 3.874162E-01 4.648994E+02 3114 LS 2.905621E+03 1.816013E+03 96 LS 1.561356E+00 1.873627E+03 3115 LS 1.171017E+04 7.318855E+03 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. FULL-SPAN MODEL SUBCASE 5 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 5.518734E-03 PITCH 2.222000E-04 URDD3 -1.917000E+00 URDD5 0.000000E+00 ELEV 3.402172E-02 SIDES 4.590776E-04 YAW 5.435000E-04 ROLL -3.222000E-05 URDD2 0.000000E+00 URDD4 0.000000E+00 URDD6 -1.099301E-28 AILERON -5.857044E-05 RUDDER -1.940631E-04 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 2.861119E-01 3.433343E+02 1000 LS 2.145839E+03 1.341150E+03

NX Nastran Aeroelastic Analysis User’s Guide

6-95

Chapter 6

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

6-96

LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS

Static Aeroelastic Analysis Sample Problems

5.328529E-02 2.301040E-02 1.913792E-02 2.264952E-01 3.838074E-02 1.761722E-02 1.684681E-02 -7.093064E-02 -6.260771E-03 -3.741469E-04 4.836466E-04 1.021406E-02 -4.882480E-03 1.086909E-04 -1.217015E-03 2.703598E-01 3.029836E-02 5.322694E-03 -3.554577E-03 2.061523E-01 6.693786E-02 1.915643E-02 4.908526E-03 1.628166E-01 7.085872E-02 3.116174E-02 1.077477E-02 1.301445E-01 6.202522E-02 3.509580E-02 1.540234E-02 1.000322E-01 4.884014E-02 3.171773E-02 1.714549E-02 6.579221E-02 3.159373E-02 2.216355E-02 1.432258E-02 2.265358E-01 3.838381E-02 1.762072E-02 1.685473E-02 2.861291E-01 5.328677E-02 2.301185E-02 1.914092E-02 6.605995E-02 3.166926E-02 2.215817E-02 1.395896E-02 1.003667E-01 4.892272E-02 3.165903E-02 1.671512E-02 1.304858E-01 6.208887E-02 3.500419E-02 1.496621E-02 1.631303E-01 7.089952E-02 3.106449E-02 1.039011E-02 2.064209E-01 6.696400E-02 1.908629E-02 4.826819E-03 2.705799E-01 3.032573E-02 5.302770E-03 -3.607364E-03 1.033240E-02 -4.811152E-03 8.433375E-05 -1.311935E-03 -7.074601E-02 -6.128056E-03 -5.604268E-04 4.032504E-04 5.245054E-05 2.913060E-04 1.869427E-04 -3.276875E-04 -1.470681E-04 3.402973E-04 2.631754E-04 -3.483750E-04 -3.725851E-04 3.075220E-04 3.081779E-04 -3.215809E-04

6.394235E+01 2.761248E+01 2.296551E+01 2.717943E+02 4.605689E+01 2.114066E+01 2.021617E+01 -8.511677E+01 -7.512926E+00 -4.489763E-01 5.803760E-01 1.225687E+01 -5.858975E+00 1.304290E-01 -1.460418E+00 3.244317E+02 3.635804E+01 6.387233E+00 -4.265492E+00 2.473827E+02 8.032543E+01 2.298771E+01 5.890231E+00 1.953799E+02 8.503046E+01 3.739408E+01 1.292973E+01 1.561734E+02 7.443027E+01 4.211496E+01 1.848281E+01 1.200386E+02 5.860817E+01 3.806128E+01 2.057459E+01 7.895065E+01 3.791247E+01 2.659625E+01 1.718709E+01 2.718430E+02 4.606057E+01 2.114486E+01 2.022567E+01 3.433549E+02 6.394413E+01 2.761422E+01 2.296910E+01 7.927194E+01 3.800311E+01 2.658980E+01 1.675075E+01 1.204400E+02 5.870726E+01 3.799084E+01 2.005814E+01 1.565829E+02 7.450665E+01 4.200503E+01 1.795945E+01 1.957563E+02 8.507942E+01 3.727739E+01 1.246813E+01 2.477051E+02 8.035680E+01 2.290354E+01 5.792183E+00 3.246959E+02 3.639087E+01 6.363324E+00 -4.328837E+00 1.239889E+01 -5.773383E+00 1.012005E-01 -1.574322E+00 -8.489522E+01 -7.353667E+00 -6.725122E-01 4.839004E-01 6.294065E-02 3.495672E-01 2.243313E-01 -3.932250E-01 -1.764817E-01 4.083568E-01 3.158105E-01 -4.180500E-01 -4.471021E-01 3.690264E-01 3.698135E-01 -3.858970E-01

NX Nastran Aeroelastic Analysis User’s Guide

1001 1002 1003 1004 1005 1006 1007 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 2000 2001 2002 2003 2004 2005 2006 2007 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111

LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS

3.996397E+02 1.725780E+02 1.435344E+02 1.698714E+03 2.878556E+02 1.321291E+02 1.263511E+02 -5.319798E+02 -4.695579E+01 -2.806102E+00 3.627350E+00 7.660545E+01 -3.661860E+01 8.151815E-01 -9.127609E+00 2.027698E+03 2.272377E+02 3.992020E+01 -2.665932E+01 1.546142E+03 5.020340E+02 1.436732E+02 3.681395E+01 1.221124E+03 5.314404E+02 2.337130E+02 8.081078E+01 9.760836E+02 4.651892E+02 2.632185E+02 1.155175E+02 7.502413E+02 3.663011E+02 2.378830E+02 1.285912E+02 4.934416E+02 2.369529E+02 1.662266E+02 1.074193E+02 1.699019E+03 2.878786E+02 1.321554E+02 1.264104E+02 2.145969E+03 3.996508E+02 1.725889E+02 1.435569E+02 4.954496E+02 2.375195E+02 1.661862E+02 1.046922E+02 7.527501E+02 3.669204E+02 2.374428E+02 1.253634E+02 9.786434E+02 4.656665E+02 2.625314E+02 1.122466E+02 1.223477E+03 5.317464E+02 2.329837E+02 7.792579E+01 1.548157E+03 5.022300E+02 1.431472E+02 3.620115E+01 2.029349E+03 2.274430E+02 3.977078E+01 -2.705524E+01 7.749303E+01 -3.608364E+01 6.325033E-01 -9.839513E+00 -5.305951E+02 -4.596042E+01 -4.203201E+00 3.024378E+00 3.933791E-01 2.184795E+00 1.402070E+00 -2.457657E+00 -1.103011E+00 2.552230E+00 1.973816E+00 -2.612812E+00 -2.794388E+00 2.306415E+00 2.311334E+00 -2.411856E+00

2.497748E+02 1.078613E+02 8.970901E+01 1.061696E+03 1.799097E+02 8.258070E+01 7.896941E+01 -3.324874E+02 -2.934737E+01 -1.753814E+00 2.267093E+00 4.787841E+01 -2.288662E+01 5.094885E-01 -5.704756E+00 1.267311E+03 1.420236E+02 2.495013E+01 -1.666208E+01 9.663388E+02 3.137712E+02 8.979576E+01 2.300871E+01 7.632026E+02 3.321502E+02 1.460706E+02 5.050674E+01 6.100522E+02 2.907432E+02 1.645115E+02 7.219846E+01 4.689008E+02 2.289382E+02 1.486768E+02 8.036951E+01 3.084010E+02 1.480956E+02 1.038916E+02 6.713708E+01 1.061887E+03 1.799241E+02 8.259712E+01 7.900652E+01 1.341230E+03 2.497817E+02 1.078680E+02 8.972305E+01 3.096560E+02 1.484497E+02 1.038664E+02 6.543263E+01 4.704688E+02 2.293253E+02 1.484018E+02 7.835211E+01 6.116521E+02 2.910417E+02 1.640822E+02 7.015411E+01 7.646732E+02 3.323416E+02 1.456148E+02 4.870362E+01 9.675981E+02 3.138937E+02 8.946697E+01 2.262572E+01 1.268343E+03 1.421519E+02 2.485674E+01 -1.690952E+01 4.843315E+01 -2.255228E+01 3.953146E-01 -6.149696E+00 -3.316219E+02 -2.872526E+01 -2.627001E+00 1.890236E+00 2.458619E-01 1.365497E+00 8.762940E-01 -1.536036E+00 -6.893816E-01 1.595144E+00 1.233635E+00 -1.633008E+00 -1.746493E+00 1.441509E+00 1.444584E+00 -1.507410E+00

Static Aeroelastic Analysis Sample Problems

93 LS -5.156262E-04 -6.187514E-01 3112 LS -3.867196E+00 94 LS 2.061618E-04 2.473941E-01 3113 LS 1.546213E+00 95 LS 3.306819E-04 3.968183E-01 3114 LS 2.480114E+00 96 LS -2.537928E-04 -3.045513E-01 3115 LS -1.903446E+00 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. FULL-SPAN MODEL SUBCASE 1 D I S P L A C E M E N T V E C T O R POINT ID. TYPE T1 T2 T3 R1 R2 90 G 0.0 0.0 0.0 0.0 0.0 97 G 0.0 0.0 -6.283592E-03 -1.375565E-12 -6.400195E-04 98 G 0.0 0.0 -8.833975E-04 -1.375565E-12 -3.400193E-04 99 G -1.190659E-19 3.067929E-18 -1.016795E-03 1.375565E-12 4.200575E-04 100 G -2.880183E-19 -4.705983E-12 -1.138205E-02 6.877825E-12 1.809385E-03 110 G 7.058997E-12 -6.304723E-13 -2.536639E-03 8.045118E-04 2.742208E-03 111 G 7.058997E-12 2.899027E-12 4.318881E-03 8.045118E-04 2.742208E-03 112 G 7.058997E-12 -4.159971E-12 -9.392160E-03 8.045118E-04 2.742208E-03 120 G 2.117699E-11 7.520553E-12 2.762073E-02 1.370994E-03 3.500449E-03 121 G 2.117699E-11 1.105005E-11 3.637186E-02 1.370994E-03 3.500449E-03 122 G 2.117699E-11 3.991054E-12 1.886961E-02 1.370994E-03 3.500449E-03 210 G -7.058998E-12 -6.304725E-13 -2.536639E-03 -8.045118E-04 2.742208E-03 211 G -7.058998E-12 2.899026E-12 4.318881E-03 -8.045118E-04 2.742208E-03 212 G -7.058998E-12 -4.159971E-12 -9.392160E-03 -8.045118E-04 2.742208E-03 220 G -2.117699E-11 7.520553E-12 2.762073E-02 -1.370994E-03 3.500449E-03 221 G -2.117699E-11 1.105005E-11 3.637186E-02 -1.370994E-03 3.500449E-03 222 G -2.117699E-11 3.991054E-12 1.886961E-02 -1.370994E-03 3.500449E-03 310 G 9.055031E-03 -4.356467E-11 -1.661024E-02 5.856540E-12 1.811563E-03 311 G 9.055031E-03 -3.495883E-11 -1.208133E-02 5.856540E-12 1.811563E-03 312 G 9.055031E-03 -5.217051E-11 -2.113914E-02 5.856540E-12 1.811563E-03 FULL-SPAN MODEL POINT ID. TYPE T1 90 G 0.0 97 G 0.0 98 G 0.0 99 G -6.025291E-19 100 G -1.536097E-18 110 G -3.036941E-04 111 G -3.036941E-04 112 G -3.036941E-04 120 G -9.110822E-04 121 G -9.110822E-04 122 G -9.110822E-04 210 G 3.036941E-04 211 G 3.036941E-04 212 G 3.036941E-04 220 G 9.110822E-04 221 G 9.110822E-04 222 G 9.110822E-04 310 G 4.300858E-02 311 G 4.300858E-02 312 G 4.300858E-02

T2 0.0 2.666585E-20 4.078307E-21 1.364871E-17 2.024627E-04 2.712499E-05 -1.247220E-04 1.789720E-04 -3.235506E-04 -4.753976E-04 -1.717035E-04 2.712499E-05 -1.247220E-04 1.789720E-04 -3.235506E-04 -4.753976E-04 -1.717035E-04 1.986964E-03 1.162923E-03 2.811004E-03

T3 0.0 -3.008065E-02 -4.220161E-03 -4.840320E-03 -5.408836E-02 -7.486501E-04 5.609721E-02 -5.759451E-02 2.282894E-01 3.304055E-01 1.261733E-01 -2.336226E-02 -1.511278E-02 -3.161174E-02 3.411017E-02 1.497908E-02 5.324126E-02 -7.892064E-02 -5.740966E-02 -1.004316E-01

V E C T O R

R1 0.0 6.577025E-05 6.577025E-05 -6.577025E-05 -3.288513E-04 3.023528E-03 3.023528E-03 3.023528E-03 5.134988E-03 5.134988E-03 5.134988E-03 -4.636744E-03 -4.636744E-03 -4.636744E-03 -7.914126E-03 -7.914126E-03 -7.914126E-03 -1.676221E-04 -1.676221E-04 -1.676221E-04

R2 0.0 -3.066049E-03 -1.626049E-03 1.998144E-03 8.593932E-03 2.273834E-02 2.273834E-02 2.273834E-02 4.084643E-02 4.084643E-02 4.084643E-02 3.299792E-03 3.299792E-03 3.299792E-03 -7.652436E-03 -7.652436E-03 -7.652436E-03 8.604390E-03 8.604390E-03 8.604390E-03

FULL-SPAN MODEL

R3 0.0 -2.635213E-21 -1.505836E-21 5.305651E-18 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 6.073881E-05 3.296162E-04 3.296162E-04 3.296162E-04

SUBCASE 3 D I S P L A C E M E N T

TYPE

T1 0.0 0.0 0.0 -5.744701E-19 -1.792114E-18 -5.542068E-05 -5.542068E-05 -5.542068E-05 -1.662620E-04 -1.662620E-04 -1.662620E-04 5.542068E-05 5.542068E-05 5.542068E-05 1.662620E-04 1.662620E-04 1.662620E-04 4.300858E-02 4.300858E-02 4.300858E-02

T2 0.0 2.666585E-20 4.078307E-21 1.421246E-17 3.694712E-05 4.949999E-06 -2.276034E-05 3.266034E-05 -5.904427E-05 -8.675460E-05 -3.133393E-05 4.949999E-06 -2.276034E-05 3.266034E-05 -5.904427E-05 -8.675460E-05 -3.133393E-05 -3.824739E-05 -2.767867E-05 -4.881611E-05

T3 0.0 -3.008065E-02 -4.220161E-03 -4.840320E-03 -5.408835E-02 -5.806080E-03 3.700949E-02 -4.862164E-02 1.850598E-01 2.554004E-01 1.147191E-01 -1.830482E-02 3.974949E-03 -4.058459E-02 7.733982E-02 8.998418E-02 6.469546E-02 -7.892063E-02 -5.740966E-02 -1.004316E-01

V E C T O R

R1 0.0 -4.769861E-06 -4.769861E-06 4.769861E-06 2.384930E-05 3.959074E-03 3.959074E-03 3.959074E-03 6.813272E-03 6.813272E-03 6.813272E-03 -3.701198E-03 -3.701198E-03 -3.701198E-03 -6.235841E-03 -6.235841E-03 -6.235841E-03 1.167260E-05 1.167260E-05 1.167260E-05

R2 0.0 -3.066049E-03 -1.626048E-03 1.998144E-03 8.593931E-03 1.712622E-02 1.712622E-02 1.712622E-02 2.813625E-02 2.813625E-02 2.813625E-02 8.911909E-03 8.911909E-03 8.911909E-03 5.057744E-03 5.057744E-03 5.057744E-03 8.604389E-03 8.604389E-03 8.604389E-03

FULL-SPAN MODEL

R3 0.0 -2.635213E-21 -1.505836E-21 5.558820E-18 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 1.108414E-05 -4.227489E-06 -4.227489E-06 -4.227489E-06

SUBCASE 4 D I S P L A C E M E N T

POINT ID. 90 G 97 G 98 G 99 G

R3 0.0 0.0 0.0 1.201939E-18 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -1.411799E-12 -3.442336E-12 -3.442336E-12 -3.442336E-12

SUBCASE 2 D I S P L A C E M E N T

POINT ID. 90 G 97 G 98 G 99 G 100 G 110 G 111 G 112 G 120 G 121 G 122 G 210 G 211 G 212 G 220 G 221 G 222 G 310 G 311 G 312 G

-2.416998E+00 9.663833E-01 1.550071E+00 -1.189654E+00

TYPE

T1 0.0 0.0 0.0 -2.503621E-07

T2 0.0 3.689642E-02 5.408786E-03 6.655405E-03

T3 0.0 -9.810241E-02 -1.262355E-02 -1.226304E-02

V E C T O R

R1 0.0 4.976837E-07 4.976837E-07 -4.976837E-07

R2 0.0 -1.027909E-02 -5.085471E-03 4.869168E-03

R3 0.0 -3.703719E-03 -2.038853E-03 2.786824E-03

NX Nastran Aeroelastic Analysis User’s Guide

6-97

Static Aeroelastic Analysis Sample Problems

Chapter 6

100 110 111 112 120 121 122 210 211 212 220 221 222 310 311 312

G G G G G G G G G G G G G G G G

-7.510864E-07 -4.260102E-02 -4.260102E-02 -4.260102E-02 -1.267980E-01 -1.267980E-01 -1.267980E-01 4.259951E-02 4.259951E-02 4.259951E-02 1.267965E-01 1.267965E-01 1.267965E-01 8.589272E-02 8.589272E-02 8.589272E-02

6.434543E-02 3.969252E-02 1.849186E-02 6.089319E-02 -8.976239E-03 -2.995897E-02 1.200649E-02 3.969252E-02 1.849186E-02 6.089319E-02 -8.976239E-03 -2.995897E-02 1.200649E-02 9.100250E-02 6.329843E-02 1.187066E-01

-1.179144E-01 -2.856540E-02 3.915555E-02 -9.628636E-02 2.952354E-01 3.839802E-01 2.064906E-01 -5.112616E-02 7.306375E-03 -1.095587E-01 1.504254E-01 2.220758E-01 7.877505E-02 -1.675051E-01 -1.245840E-01 -2.104261E-01

-2.488418E-06 9.215011E-03 9.215011E-03 9.215011E-03 1.627373E-02 1.627373E-02 1.627373E-02 -3.131837E-03 -3.131837E-03 -3.131837E-03 -5.904732E-03 -5.904732E-03 -5.904732E-03 8.392542E-04 8.392542E-04 8.392542E-04

FULL-SPAN MODEL

ID. G G G G G G G G G G G G G G G G G G G G

8.572550E-03 8.480265E-03 8.480265E-03 8.480265E-03 8.393090E-03 8.393090E-03 8.393090E-03 8.480265E-03 8.480265E-03 8.480265E-03 8.393090E-03 8.393090E-03 8.393090E-03 1.108163E-02 1.108163E-02 1.108163E-02

SUBCASE 5 D I S P L A C E M E N T

POINT 90 97 98 99 100 110 111 112 120 121 122 210 211 212 220 221 222 310 311 312

1.719605E-02 2.708838E-02 2.708838E-02 2.708838E-02 3.549792E-02 3.549792E-02 3.549792E-02 2.337301E-02 2.337301E-02 2.337301E-02 2.866016E-02 2.866016E-02 2.866016E-02 1.716842E-02 1.716842E-02 1.716842E-02

V E C T O R

TYPE T1 T2 T3 R1 R2 R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.351860E-29 -1.201605E-02 -8.181624E-09 -1.224698E-03 -1.335956E-30 0.0 2.067551E-30 -1.686073E-03 -8.181624E-09 -6.495971E-04 -7.634034E-31 -2.333323E-19 5.878382E-18 -1.934396E-03 8.181624E-09 7.985907E-04 2.304321E-18 -5.120325E-19 -4.141365E-07 -2.161905E-02 4.090812E-08 3.435126E-03 -1.242409E-07 6.212047E-07 -5.548402E-08 -4.819604E-03 1.530897E-03 5.202882E-03 -1.242409E-07 6.212047E-07 2.551183E-07 8.187599E-03 1.530897E-03 5.202882E-03 -1.242409E-07 6.212047E-07 -3.660864E-07 -1.782681E-02 1.530897E-03 5.202882E-03 -1.242409E-07 1.863614E-06 6.618211E-07 5.243254E-02 2.607960E-03 6.632013E-03 -1.242409E-07 1.863614E-06 9.724234E-07 6.901257E-02 2.607960E-03 6.632013E-03 -1.242409E-07 1.863614E-06 3.512187E-07 3.585251E-02 2.607960E-03 6.632013E-03 -1.242409E-07 -6.212047E-07 -5.548402E-08 -4.817418E-03 -1.530448E-03 5.205313E-03 -1.242409E-07 -6.212047E-07 2.551183E-07 8.195865E-03 -1.530448E-03 5.205313E-03 -1.242409E-07 -6.212047E-07 -3.660864E-07 -1.783070E-02 -1.530448E-03 5.205313E-03 -1.242409E-07 -1.863614E-06 6.618211E-07 5.245187E-02 -2.607139E-03 6.638009E-03 -1.242409E-07 -1.863614E-06 9.724234E-07 6.904689E-02 -2.607139E-03 6.638009E-03 -1.242409E-07 -1.863614E-06 3.512187E-07 3.585685E-02 -2.607139E-03 6.638009E-03 -1.242409E-07 1.719117E-02 -9.694116E-07 -3.154488E-02 -1.449587E-07 3.439302E-03 -4.514883E-07 1.719117E-02 1.593091E-07 -2.294662E-02 -1.449587E-07 3.439302E-03 -4.514883E-07 1.719117E-02 -2.098132E-06 -4.014314E-02 -1.449587E-07 3.439302E-03 -4.514883E-07

FULL-SPAN MODEL F O ELEMENT BEND-MOMENT END-A ID. PLANE 1 PLANE 2 101 -9.094947E-13 0.0 10 -3.000000E+04 0.0 103 -3.800382E+04 -1.103879E-10 104 -4.600763E+04 -9.740170E-11 110 6.226683E+04 2.773295E-12 120 3.299180E+04 1.688287E-12 210 6.226683E+04 -5.611514E-11 220 3.299179E+04 1.485610E-12 310 -2.386749E+02 7.377408E-05

SUBCASE 1 R C E S I N B A R E L E M E N T S ( C B BEND-MOMENT END-B - SHEAR PLANE 1 PLANE 2 PLANE 1 PLANE 2 -3.000000E+04 0.0 3.000000E+03 0.0 -3.800382E+04 0.0 1.600763E+03 0.0 -4.600763E+04 -9.730721E-11 1.600763E+03-2.616130E-12 -9.292499E+04 1.219797E-04 4.691736E+03-1.219798E-05 3.846394E+04 1.742006E-12 4.122783E+03 1.786245E-13 4.667687E+03 7.875740E-14 2.452940E+03 1.393894E-13 3.846394E+04 1.703835E-12 4.122783E+03-1.001454E-11 4.667683E+03 5.972845E-14 2.452940E+03 1.234850E-13 4.999999E+01 5.672844E-05-4.999997E+01 2.952393E-06

FULL-SPAN MODEL

A R ) AXIAL FORCE TORQUE 0.0 1.572822E-16 0.0 1.485610E-04 -1.371640E-10 1.485610E-04 -9.731654E-11 2.971220E-04 -3.755793E-14 1.756271E+04 -1.877892E-14 8.084671E+03 -6.691022E-12-1.756271E+04 1.829143E-14-8.084670E+03 -8.660255E+01-9.825661E-05

SUBCASE 2

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 101 0.0 -6.310887E-30 -1.440000E+05 -9.757820E-14 1.440000E+04 9.757820E-15 0.0 0.0 102 -1.440000E+05 -9.757820E-14 -1.812095E+05 -1.626303E-13 7.441901E+03 1.301043E-14 0.0 -7.103188E+03 103 -1.812095E+05 -4.982818E-10 -2.184190E+05 -4.185348E-10 7.441901E+03 -1.594940E-11 -6.941134E-10-7.103188E+03 104 -2.184190E+05 -4.175447E-10 -4.411594E+05 -5.247833E+03 2.227404E+04 5.247834E+02 -5.377354E-10-1.420638E+04 110 4.939954E+05 2.364686E-11 3.699034E+05 2.889783E-11 2.149338E+04 -9.094947E-13 3.637979E-12 4.578346E+05 120 2.884018E+05 2.364686E-11 1.828292E+05 7.893954E-12 9.142859E+03 1.364242E-12 -9.094947E-13 3.166696E+05 210 9.845001E+04 -2.546585E-10 -4.600900E+03 6.565060E-11 1.784894E+04 -5.547918E-11 -7.821654E-11 2.917989E+05 220 2.482422E+04 3.092282E-11 -1.390450E+05 7.293457E-12 1.419149E+04 2.046363E-12 9.094947E-13 2.408330E+05 310 -1.145640E+03 1.301454E+04 2.400000E+02 -7.836728E+03 -2.399999E+02 3.611547E+03 -4.156923E+02 1.357362E+04 FULL-SPAN MODEL F O R C E S I N B A R E L E M E N T S ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 101 -7.275958E-12 -6.310887E-30 -1.440000E+05 -9.757820E-14 1.440000E+04 102 -1.440000E+05 -9.757820E-14 -1.812095E+05 -1.626303E-13 7.441897E+03 103 -1.812095E+05 -5.129836E-10 -2.184190E+05 -4.475806E-10 7.441897E+03 104 -2.184190E+05 -4.507738E-10 -4.411593E+05 -9.576694E+02 2.227404E+04 110 3.863516E+05 1.637090E-11 2.782490E+05 -1.316580E-11 1.872393E+04 120 2.440580E+05 -2.091838E-11 1.013483E+05 5.336470E-12 1.235902E+04 210 2.060938E+05 -4.908998E-10 8.705350E+04 1.851625E-10 2.061839E+04 220 6.916809E+04 -3.137757E-11 -5.756406E+04 4.722849E-12 1.097533E+04 310 -1.145640E+03 -3.366487E+03 2.400000E+02 4.837149E+02-2.399999E+02

6-98

NX Nastran Aeroelastic Analysis User’s Guide

SUBCASE 3 ( C B A R ) - SHEAR AXIAL PLANE 2 FORCE TORQUE 9.757820E-15 0.0 0.0 1.301043E-14 0.0 5.151450E+02 -1.308061E-11 -6.617895E-10 5.151450E+02 9.576694E+01 -7.013626E-10 1.030290E+03 5.115908E-12 4.342837E-11 2.347611E+05 -2.273737E-12 2.148681E-11 1.755404E+05 -1.170974E-10 -1.305125E-10 6.872545E+04 -3.126388E-12 -2.137313E-11 9.970392E+04 -6.668746E+02 -4.156923E+02-8.378195E+02

Static Aeroelastic Analysis Sample Problems

FULL-SPAN MODEL

SUBCASE 4

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 101 0.0 0.0 -5.193619E+05 -1.438445E+05 5.193619E+04 1.438445E+04 0.0 -8.881784E-16 102 -5.193619E+05 -1.438445E+05 -4.977316E+05 -2.084692E+05 -4.326061E+03 1.292495E+04 0.0 -5.374984E+01 103 -4.977316E+05 -2.084692E+05 -4.761013E+05 -2.730940E+05 -4.326061E+03 1.292495E+04 -2.884171E+02-5.374984E+01 104 -4.761013E+05 -2.730940E+05 -7.565864E+05 -2.267928E+05 2.804851E+04 -4.630117E+03 -2.884171E+02-1.074997E+02 110 6.677701E+05 5.011036E+04 4.518922E+05 4.196039E+04 3.739116E+04 1.411615E+03 -1.868028E+04 1.713980E+05 120 3.998546E+05 4.342184E+04 4.691892E+04 6.313428E+01 3.056513E+04 3.754975E+03 -9.340141E+03 8.126596E+04 210 2.689082E+05 5.011036E+04 2.332628E+05 4.196039E+04 6.173956E+03 1.411615E+03 1.868028E+04-1.638838E+05 220 1.785490E+05 4.342184E+04 3.990443E+04 6.313428E+01 1.200698E+04 3.754975E+03 9.340141E+03-6.911652E+04 310 2.115105E+03 -4.594926E+05 2.776144E+02 -6.484483E+04 3.182628E+02 -6.835501E+04 -2.558681E+01 1.123146E+05 FULL-SPAN MODEL

SUBCASE 5

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 101 -3.637979E-12 -5.877472E-39 -5.751000E+04 -4.946854E-23 5.751000E+03 4.946854E-24 0.0 3.122502E-16 102 -5.751000E+04 -4.946854E-23 -7.240934E+04 -8.244757E-23 2.979869E+03 6.595806E-24 0.0 8.836154E-01 103 -7.240934E+04 -2.112840E-10 -8.730869E+04 -1.869028E-10 2.979869E+03 -4.876237E-12 -2.687988E-10 8.836154E-01 104 -8.730869E+04 -1.855495E-10 -1.763447E+05 1.073442E+01 8.903599E+03 -1.073442E+00 -1.605313E-10 1.767231E+00 110 1.183698E+05 5.293543E-12 7.299042E+04 3.319302E-12 7.859931E+03 3.419487E-13 -7.460699E-14 3.314685E+04 120 6.259227E+04 3.204548E-12 8.739168E+03 1.380635E-13 4.663815E+03 2.655653E-13 -3.552714E-14 1.513669E+04 210 1.184088E+05 -7.928591E-11 7.302909E+04 -2.660700E-11 7.859997E+03 -9.124257E-12 -2.545875E-11-3.324602E+04 220 6.261458E+04 2.838618E-12 8.780066E+03 1.823660E-13 4.662206E+03 2.300382E-13 3.197442E-14-1.520753E+04 310 -4.575398E+02 -6.756530E+00 9.585001E+01 9.408445E+00 -9.584995E+01 -2.799856E+00 -1.660171E+02-1.629592E+01 FULL-SPAN MODEL SUBCASE 1 ELEMENT SA1 ID. SB1 101 2.619346E-12 8.640005E+04 102 8.640005E+04 1.094511E+05 103 1.094511E+05 1.325021E+05 104 1.325021E+05 2.676242E+05 110 -1.793286E+05 -1.107762E+05 120 -9.501644E+04 -1.344295E+04 210 -1.793286E+05 -1.107762E+05 220 -9.501642E+04 -1.344294E+04 310 6.873841E+02 -1.440002E+02

S T R E S S E S I N SA2 SA3 SB2 SB3 2.619346E-12 -2.619346E-12 8.640005E+04 -8.640005E+04 8.640005E+04 -8.640005E+04 1.094511E+05 -1.094511E+05 1.094511E+05 -1.094511E+05 1.325021E+05 -1.325021E+05 1.325021E+05 -1.325021E+05 2.676242E+05 -2.676242E+05 -1.793286E+05 1.793286E+05 -1.107762E+05 1.107762E+05 -9.501644E+04 9.501644E+04 -1.344295E+04 1.344295E+04 -1.793286E+05 1.793286E+05 -1.107762E+05 1.107762E+05 -9.501642E+04 9.501642E+04 -1.344294E+04 1.344294E+04 6.873843E+02 -6.873843E+02 -1.440000E+02 1.440000E+02

B A R E L E M E N T S SA4 AXIAL SB4 STRESS -2.619346E-12 0.0 -8.640005E+04 -8.640005E+04 0.0 -1.094511E+05 -1.094511E+05 -3.429099E-11 -1.325021E+05 -1.325021E+05 -2.432913E-11 -2.676242E+05 1.793286E+05 -2.503862E-14 1.107762E+05 9.501644E+04 -1.251928E-14 1.344295E+04 1.793286E+05 -4.460681E-12 1.107762E+05 9.501642E+04 1.219428E-14 1.344294E+04 -6.873841E+02 -5.773503E+01 1.440002E+02

FULL-SPAN MODEL ELEMENT SA1 ID. SB1 101 2.103629E-29 4.147203E+05 102 4.147203E+05 5.218837E+05 103 5.218837E+05 6.290471E+05 104 6.290472E+05 1.288033E+06 110 -1.422708E+06 -1.065322E+06 120 -8.305978E+05 -5.265484E+05 210 -2.835362E+05 1.325060E+04 220 -7.149380E+04 4.004497E+05 310 -1.622237E+04 1.106389E+04 FULL-SPAN MODEL ELEMENT SA1 ID. SB1 101 2.095477E-11 4.147203E+05 102 4.147203E+05 5.218837E+05 103 5.218837E+05 6.290471E+05 104 6.290471E+05 1.273732E+06 110 -1.112693E+06 -8.013576E+05 120 -7.028874E+05 -2.918832E+05

( C B A R ) SA-MAX SB-MAX 2.619346E-12 8.640005E+04 8.640005E+04 1.094511E+05 1.094511E+05 1.325021E+05 1.325021E+05 2.676242E+05 1.793286E+05 1.107762E+05 9.501644E+04 1.344295E+04 1.793286E+05 1.107762E+05 9.501642E+04 1.344294E+04 6.296493E+02 8.626512E+01

SA-MIN M.S.-T SB-MIN M.S.-C -2.619346E-12 -8.640005E+04 -8.640005E+04 -1.094511E+05 -1.094511E+05 -1.325021E+05 -1.325021E+05 -2.676242E+05 -1.793286E+05 -1.107762E+05 -9.501644E+04 -1.344295E+04 -1.793286E+05 -1.107762E+05 -9.501642E+04 -1.344294E+04 -7.451194E+02 -2.017352E+02

SUBCASE 2 S T R E S SA2 SB2 -2.103629E-29 4.147203E+05 4.147203E+05 5.218837E+05 5.218837E+05 6.290471E+05 6.290472E+05 1.253047E+06 -1.422708E+06 -1.065322E+06 -8.305978E+05 -5.265484E+05 -2.835362E+05 1.325060E+04 -7.149380E+04 4.004497E+05 2.282126E+04 -1.244629E+04

S E S I N B A R E L E M SA3 SA4 SB3 SB4 2.103629E-29 -2.103629E-29 -4.147203E+05 -4.147203E+05 -4.147203E+05 -4.147203E+05 -5.218837E+05 -5.218837E+05 -5.218837E+05 -5.218837E+05 -6.290471E+05 -6.290471E+05 -6.290472E+05 -6.290472E+05 -1.253047E+06 -1.288033E+06 1.422708E+06 1.422708E+06 1.065322E+06 1.065322E+06 8.305978E+05 8.305978E+05 5.265484E+05 5.265484E+05 2.835362E+05 2.835362E+05 -1.325060E+04 -1.325060E+04 7.149380E+04 7.149380E+04 -4.004497E+05 -4.004497E+05 -2.282126E+04 1.622237E+04 1.244629E+04 -1.106389E+04

S T R E S S E S I N SA2 SA3 SB2 SB3 2.095477E-11 -2.095477E-11 4.147203E+05 -4.147203E+05 4.147203E+05 -4.147203E+05 5.218837E+05 -5.218837E+05 5.218837E+05 -5.218837E+05 6.290471E+05 -6.290471E+05 6.290471E+05 -6.290471E+05 1.267348E+06 -1.267348E+06 -1.112693E+06 1.112693E+06 -8.013576E+05 8.013576E+05 -7.028874E+05 7.028874E+05 -2.918832E+05 2.918832E+05

E N T S AXIAL STRESS 0.0 0.0 -1.735284E-10 -1.344338E-10 2.425319E-12 -6.063298E-13 -5.214436E-11 6.063298E-13 -2.771282E+02

B A R E L E M E N T S SA4 AXIAL SB4 STRESS -2.095477E-11 0.0 -4.147203E+05 -4.147203E+05 0.0 -5.218837E+05 -5.218837E+05 -1.654474E-10 -6.290471E+05 -6.290471E+05 -1.753407E-10 -1.273732E+06 1.112693E+06 2.895225E-11 8.013576E+05 7.028874E+05 1.432454E-11 2.918832E+05

( C B A R ) SA-MAX SA-MIN M.S.-T SB-MAX SB-MIN M.S.-C 2.103629E-29 -2.103629E-29 4.147203E+05 -4.147203E+05 4.147203E+05 -4.147203E+05 5.218837E+05 -5.218837E+05 5.218837E+05 -5.218837E+05 6.290471E+05 -6.290471E+05 6.290472E+05 -6.290472E+05 1.288033E+06 -1.288033E+06 1.422708E+06 -1.422708E+06 1.065322E+06 -1.065322E+06 8.305978E+05 -8.305978E+05 5.265484E+05 -5.265484E+05 2.835362E+05 -2.835362E+05 1.325060E+04 -1.325060E+04 7.149380E+04 -7.149380E+04 4.004497E+05 -4.004497E+05 2.254413E+04 -2.309839E+04 1.216917E+04 -1.272342E+04 SUBCASE 3 ( C B A R ) SA-MAX SA-MIN M.S.-T SB-MAX SB-MIN M.S.-C 2.095477E-11 -2.095477E-11 4.147203E+05 -4.147203E+05 4.147203E+05 -4.147203E+05 5.218837E+05 -5.218837E+05 5.218837E+05 -5.218837E+05 6.290471E+05 -6.290471E+05 6.290471E+05 -6.290471E+05 1.273732E+06 -1.273732E+06 1.112693E+06 -1.112693E+06 8.013576E+05 -8.013576E+05 7.028874E+05 -7.028874E+05 2.918832E+05 -2.918832E+05

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-5.935506E+05 -2.507142E+05 -1.992042E+05 1.657846E+05 8.349175E+03 -1.416773E+03

-5.935506E+05 -2.507142E+05 -1.992042E+05 1.657846E+05 -1.750286E+03 3.437187E+01

5.935506E+05 2.507142E+05 1.992042E+05 -1.657846E+05 1.750286E+03 -3.437187E+01

5.935506E+05 2.507142E+05 1.992042E+05 -1.657846E+05 -8.349175E+03 1.416773E+03

-8.700832E-11 -1.424875E-11 -2.771282E+02

FULL-SPAN MODEL ELEMENT SA1 ID. SB1 101 0.0 1.975245E+06 102 1.975245E+06 2.128365E+06 103 2.128365E+06 2.281486E+06 104 2.281486E+06 2.934946E+06 110 -1.998345E+06 -1.364391E+06 120 -1.216715E+06 -1.352213E+05 210 -8.496216E+05 -7.347380E+05 220 -5.793543E+05 -1.150195E+05 310 6.831474E+05 9.646771E+04

5.935506E+05 2.507142E+05 1.992042E+05 1.657846E+05 8.072047E+03 1.139645E+03

-5.935506E+05 -2.507142E+05 -1.992042E+05 -1.657846E+05 -8.626303E+03 -1.693901E+03

SUBCASE 4 S T R E S S E S I N SA2 SA3 SB2 SB3 0.0 0.0 1.016282E+06 -1.016282E+06 1.016282E+06 -1.016282E+06 7.385705E+05 -7.385705E+05 7.385705E+05 -7.385705E+05 4.608593E+05 -4.608593E+05 4.608593E+05 -4.608593E+05 1.422994E+06 -1.422994E+06 -1.848014E+06 1.848014E+06 -1.238510E+06 1.238510E+06 -1.086449E+06 1.086449E+06 -1.350319E+05 1.350319E+05 -6.992906E+05 6.992906E+05 -6.088568E+05 6.088568E+05 -4.490888E+05 4.490888E+05 -1.148301E+05 1.148301E+05 -6.953304E+05 6.953304E+05 -9.806677E+04 9.806677E+04

FULL-SPAN MODEL

B A R E L E M E N T S SA4 AXIAL SB4 STRESS 0.0 0.0 -1.975245E+06 -1.975245E+06 0.0 -2.128365E+06 -2.128365E+06 -7.210429E+01 -2.281486E+06 -2.281486E+06 -7.210429E+01 -2.934946E+06 1.998345E+06 -1.245352E+04 1.364391E+06 1.216715E+06 -6.226760E+03 1.352213E+05 8.496216E+05 1.245352E+04 7.347380E+05 5.793543E+05 6.226760E+03 1.150195E+05 -6.831474E+05 -1.705788E+01 -9.646771E+04

( C B A R ) SA-MAX SA-MIN M.S.-T SB-MAX SB-MIN M.S.-C 0.0 0.0 1.975245E+06 -1.975245E+06 1.975245E+06 -1.975245E+06 2.128365E+06 -2.128365E+06 2.128293E+06 -2.128437E+06 2.281414E+06 -2.281558E+06 2.281414E+06 -2.281558E+06 2.934874E+06 -2.935018E+06 1.985891E+06 -2.010798E+06 1.351937E+06 -1.376844E+06 1.210488E+06 -1.222942E+06 1.289945E+05 -1.414480E+05 8.620751E+05 -8.371681E+05 7.471915E+05 -7.222845E+05 5.855811E+05 -5.731276E+05 1.212463E+05 -1.087928E+05 6.953133E+05 -6.953474E+05 9.804972E+04 -9.808383E+04 SUBCASE 5

S T R E S S E S I N B A R E L E M E N T S ( C B A R ) ELEMENT SA1 SA2 SA3 SA4 AXIAL SA-MAX SA-MIN M.S.-T ID. SB1 SB2 SB3 SB4 STRESS SB-MAX SB-MIN M.S.-C 101 1.047739E-11 1.047739E-11 -1.047739E-11 -1.047739E-11 0.0 1.047739E-11 -1.047739E-11 1.656289E+05 1.656289E+05 -1.656289E+05 -1.656289E+05 1.656289E+05 -1.656289E+05 102 1.656289E+05 1.656289E+05 -1.656289E+05 -1.656289E+05 0.0 1.656289E+05 -1.656289E+05 2.085390E+05 2.085390E+05 -2.085390E+05 -2.085390E+05 2.085390E+05 -2.085390E+05 103 2.085390E+05 2.085390E+05 -2.085390E+05 -2.085390E+05 -6.719971E-11 2.085390E+05 -2.085390E+05 2.514492E+05 2.514492E+05 -2.514492E+05 -2.514492E+05 2.514492E+05 -2.514492E+05 104 2.514492E+05 2.514492E+05 -2.514492E+05 -2.514492E+05 -4.013282E-11 2.514492E+05 -2.514492E+05 5.078372E+05 5.079088E+05 -5.079088E+05 -5.078372E+05 5.079088E+05 -5.079088E+05 110-3.409051E+05-3.409051E+05 3.409051E+05 3.409051E+05 -4.973799E-14 3.409051E+05 -3.409051E+05 -2.102125E+05-2.102125E+05 2.102125E+05 2.102125E+05 2.102125E+05 -2.102125E+05 120-1.802658E+05-1.802658E+05 1.802658E+05 1.802658E+05 -2.368476E-14 1.802658E+05 -1.802658E+05 -2.516882E+04-2.516882E+04 2.516882E+04 2.516882E+04 2.516882E+04 -2.516882E+04 210-3.410176E+05-3.410176E+05 3.410176E+05 3.410176E+05 -1.697250E-11 3.410176E+05 -3.410176E+05 -2.103239E+05-2.103239E+05 2.103239E+05 2.103239E+05 2.103239E+05 -2.103239E+05 22 -1.803301E+05-1.803301E+05 1.803301E+05 1.803301E+05 2.131628E-14 1.803301E+05 -1.803301E+05 -2.528661E+04-2.528661E+04 2.528661E+04 2.528661E+04 2.528661E+04 -2.528661E+04 310 1.327850E+03 1.307581E+03 -1.307581E+03 -1.327850E+03 -1.106781E+02 1.217172E+03 -1.438528E+03 -2.901609E+02 -2.619355E+02 2.619355E+02 2.901609E+02 1.794828E+02 -4.008390E+02

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Listing 6-22. Output for FSW Airplane in Unsymmetric Maneuvers

6.7

FSW Airplane with Bodies (Example HA144F)

This example is a modification of the full-span Example HA144E to include a fuselage, under-wing pylon-mounted stores as well as wing incidence and outboard dihedral. Three views are shown in Figure 6-7. The primary purpose of this example is to illustrate the inclusion of body interference in a quasi-static problem. However, a tapered fuselage is also chosen to demonstrate a modeling feature of the Giesing Method of Images that requires moving the canard vertically (in this example) from its physical location so that its root intersects the interference tube while maintaining the same span and spanwise location. The DMI input for wing incidence is affected by the dihedral, and this feature is also illustrated. Only the sea-level flight condition at m = 0.9 is considered since the example is only intended to illustrate the data input.

The Configuration The fuselage is assumed to be 40.0 ft long with the nose 5.0 ft forward of GRID 97 and the aft end 5.0 ft behind GRID 100. Its assumed circular cross section has a diameter of 5.0 ft at the root of the wing and tapers in diameter to 4.0 ft at the root of the canard; a parabolic ogive extends forward of the canard to the nose. The wing and canard spans are maintained at 40.0 and 10.0 ft, respectively, as before. The wing stores are mounted on pylons at the midspan of each wing. The pylons have a chord of 5.0 ft and extend below the wing 1.0 ft and are centrally aligned on the wing chord. Each store is 10.0 ft long with a diameter of 1.0 ft in the region of the pylon with a pointed nose and an aft end. The stores are centrally aligned along the pylons. The wings are assumed to have an incidence of 0.1 deg relative to the fuselage centerline and to have a dihedral break at the pylons such that the wing tip chord is 2.0 ft above the inboard wing plane. Structural models for the canard, pylons, and stores are added. An elastic axis (CBAR) representation of the canard is assumed from the fuselage centerline (GRID 90) to the canard tips along the canard midchord (50%) line. A vertical elastic axis is assumed for the pylons between the wing elastic axis and the store elastic axis, which is assumed along the store centerline. The same stiffness properties are assumed for the canard, pylon, and store elastic axes as for the wing.

Additional Structural Model The additions to the structural model begin with grid points. First, the wings have been modified to introduce the dihedral: GRIDs 115 and 215 are introduced on the right and left wing elastic axes, respectively, to define the pylon station and dihedral break. Next, the points on the canard elastic axis are GRIDs 88, 89, 91, 92, 890, and 910. GRIDs 88 and 92 are at the left and right canard tips, respectively. GRIDS 89 and 91 are at the left and right sides of the fuselage, respectively. GRIDs 890 and 910 are also at the sides of the fuselage, coincident with GRIDs 89 and 91, respectively, and are introduced by the requirements of the aerodynamic Method of Images, as is discussed below. The canard hinge line is now assumed to be located along its elastic axis (in previous examples with this configuration, the hinge line was assumed to be located at the canard quarter-chord). GRIDs 150 and 151 locate the right pylon/store: 150 is forward on the store elastic axis, and 151 is aft on the store elastic axis and also at the bottom of the pylon elastic axis. The corresponding GRIDs on the left pylon/store are 250 and 251.

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Figure 6-7. Three Views of the Aerodynamic Model for the FSW Airplane with Fuselage and Stores The wing elastic axes are represented by two BAR elements on each side. On the right wing, CBAR 115 connects GRIDs 110 and 115, and CBAR 120 connects GRIDs 115 and 120. On the left wing, CBAR 215 connects GRIDs 210 and 215, and CBAR 220 connects GRIDs 215 and 220. The canard structural elements also consist of BARs: CBAR 89 between GRIDs 88 and 89, CBAR 90 between GRIDs 890 and 90, CBAR 91 between GRIDs 90 and 910, and CBAR 92 between GRIDs 91 and 92. The right pylon structural element is CBAR 151 between GRIDs 115 and 150, and the right store structural element is CBAR 150 between GRIDs 150 and 151. The corresponding left pylon/store elements are CBARs 251 and 250. All of these new elements refer to the property and material entries for the wing, PBAR 101 and MAT1 1, for their stiffnesses. No additional weights are considered in the additions of the canard structure and the pylons and stores.

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Aerodynamic Model The wings are divided into two panels on each side, having the same size boxes as before, with the fuselage replacing the inboard strips of boxes on each side, and also with an outboard dihedral angle (11.31 deg). The right inboard wing is specified on lifting surface entry CAERO1 1104, which refers to PAERO1 1000 for the identification of any associated bodies. PAERO1 1000 now refers to the fuselage CAERO2 4000, the right store CAERO2 4510, and the left store, CAERO2 4610. CAERO1 1104 then specifies 3 equal span strips and 4 equal chord boxes in the first interference group. The planform geometry on the continuation entry begins with the leading edge location at the root of the wing (which is now displaced by the radius of the fuselage in the wing region, 2.5 ft), the root chord length, and the leading edge location at the dihedral break and its chord length. The right outboard wing is specified on entry CAERO1 1116, which associates the panel with the three bodies using the PAERO1 entry and divides the panel equally into four spanwise strips and four chordwise boxes in the first interference group. The continuation entry provides the leading edge location and chord length at the dihedral break and then the leading edge location and chord length at the tip. Two splines are now needed on each wing because of the dihedral break. On the right inboard wing, SPLINE2 1602 with its axis CORD2R 2 as before, but now with grid points designed by SETG 1101. On the right outboard wing, the spline is prescribed by SPLINE2 1603 with a new axis CORD2R 3 and grid points designated by SETG 1102 prescribe the spline. CAERO1s 2100 and 2116 provide similar data for the left wing, arbitrarily numbering the boxes from the tip inboard, whereas the right wing box numbering scheme begins at the root and increases outboard. The left wing spline and spline axes are also similar: for the outboard wing, SPLINE2 2601 with CORD2R 30 and SETG 2102, and for the inboard wing, SPLINE2 2602 with CORD2R 20 and SETG 2101.

Pylons The right wing pylon is specified on lifting surface entry CAERO1 3500 and is modelled as one strip with two chordwise boxes and is associated with all three bodies. Its spline is SPLINE2 4520 with axis CORD2R 450 and structural connection to grid points SETG 4521. The left wing pylon is similar to the lifting surface entry CAERO1 3600; its spline is SPLINE2 4620 with axis CORD2R 460 and structural connection to points SETG 4621.

Canard The right side of the canard is specified on entry CAERO1 1000. It associates the panel with the three bodies, specifies the right side geometry in the Cartesian coordinate system CORD2R 900 (for reasons to be discussed below regarding the requirements of the Method of Images), divides the span unequally into two strips with four chordwise boxes via AEFACT 1000 in order to align the trailing vortices with the wing in the same interference group. The geometry on the continuation entry begins with the leading edge location at the root of the canard (displaced here by the radius of the fuselage in the canard region, 2.0 ft), the root chord length, and then the leading edge location of the tip and the tip chord. The right canard is now interconnected using the new SPLINE2 1502 with its elastic axis along CORD2R 90 to the structural grid points on both sides, SET1 1000. The hinge line is also aligned with the elastic axis in this modification to Example HA144E. CAERO1 2000 and AEFACT 2000 provide similar data for the left side of the canard but with the box numbering scheme beginning at the tip leading edge. The associated spline is SPLINE2 2501 with the same axis CORD2R 90 and structural connection SET1 1000.

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Fin The fin is again specified on entry CAERO1 3100 but with only three strips because of the presence of the fuselage and modified geometry on the continuation entry for the leading edge location at the fuselage junction.

Fuselage The fuselage aerodynamic properties are given on entry CAERO2 4000: •

In the PID field, the reference is to the entry PAERO2 4020, which specifies additional data for the fuselage.

•

The CP field refers to the coordinate system for the fuselage geometry; in this case, it is blank and thus refers to the basic system.

•

The NSB field specifies eight equal length divisions of the fuselage.

•

Because of the NSB field specification, the LSB field is not needed.

•

The NINT field also specifies eight equal divisions of the interference tube.

•

The LINT field, like the LSB field, is again not needed.

•

The IGID field specifies the same interference group as the other aerodynamic components.

The continuation entry contains the coordinates of the fuselage nose and its length. The additional fuselage data on entry PAERO2 4020 include: •

The ORIENT field, which is the body orientation, (in this case ZY) specifies that the fuselage can move both vertically and laterally.

•

The WIDTH field, which is actually the semiwidth or radius of 2.5 ft in this case since the cross section is circular. This is the maximum width of the fuselage at the root of the wing, and this also determines the width of the interference tube.

•

The AR field, is the aspect ratio (the ratio of the height to the width) of the possibly elliptical cross section; in this case, AR = 1.0 specifies a circular section.

•

The LRSB field, refers to the entry AEFACT 4015, which lists the half-widths at the nine end points of the eight slender body elements.

•

The LRIB field is the reference to a list of slender body half-widths at the end points of the eight interference elements but is left blank in accordance with the recommendation in Aerodynamic Theories.

•

The LTH1 field refers to the entry AEFACT 4018, which lists the angular positions (the q1 array) around the periphery of the interference tube for averaging the interfering flow from the lifting surfaces and other slender bodies; in this case, four sampling points are selected beginning at 45 deg and every 90 deg beyond.

•

The LTH2 field could refer to a alternate list of angular positions, as discussed in Aerodynamic Theories (the θ2 array), but is not used here.

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The continuation entry lists pairs of first and last interference elements that use the array; in this case, all eight interference elements use the same array. The fuselage spline is SPLINE2 4000 and is connected to structural grid points SET1 4001. Note that the linear spline used for a body does not require the specification of a coordinate system for the spline axis along the body. The right store aerodynamic properties are entered on CAERO2 4510. The third field of the entry refers to PAERO2 4520 for additional body data. The fifth and sixth fields specify four equal length divisions of both the body and its interference tube. The continuation locates the store nose below the wing leading edge and gives its length as 10.0 ft. The additional properties on PAERO2 4520 include the ZY degrees of freedom, the reference width of 0.5 ft, and AR = 1.0 for the circular cross section. LRSB refers to AEFACT 4515 for the list of store half-widths at the five end points of the four slender body elements, and LTH1 refers to the same q1 -array used for the fuselage, AEFACT 4018. The continuation entry specifies that the four interference elements will use the single array. The right store spline is SPLINE2 4525 and is interconnected to grids SET1 4525. The left store CAERO2 4610 entry is similar except for its nose location and uses the additional data for the right store on entry PAERO2 4520. Its spline is SPLINE2 4625 with connections to SET1 4625.

Wing Incidence and Dihedral The remaining new configuration data are the DMI entries to specify the wing incidence of 0.1 degrees = 0.0017453 radians. Because of the dihedral angle of = tan−10.2 = 11.31 deg, the outboard wing panels have less incidence than the inboard panels by a factor of cos G, and the outboard value becomes 0.0017114 rad. The DMI name for incidence, twist, and camber is W2GJ. Since the incidence is constant for a number of boxes in this example, the DMI input can use the convenience of the "THRU" data input option, but it requires the j-set numbering of the aerodynamic boxes in the sequence from the right wing root to the left wing root (see Aerodynamic Theories ). The numbering begins with the lowest numbered CAERO1 identification number. The right side canard (CAERO1 1000) has eight boxes; thus, the first inboard wing (CAERO1 1104) box is No. 9; the last box on the right inboard wing is No. 20 since there are 12 boxes on the inboard wing. There are 16 boxes on the right outboard wing (CAERO1 1116), and their numbers range from No. 21 to No. 36. The left canard (CAERO1 2000) intervenes in the sequence with its eight boxes; therefore, the next 16 wing boxes begin with No. 45 at the outboard left wing (CAERO1 2100) tip and range to No. 60 at the dihedral break. The twelve inboard left wing (CAERO1 2116) boxes range from No. 61 to No. 72. The remaining lifting surfaces are beyond the range of concern to DMI W2GJ (and the bodies must have even higher numbers). However, it is necessary to specify the correct number of rows for the matrix which is the M field (152 in this case, see Aerodynamic Modeling ) on the parent DMI entry.

Interference The final modeling task is to satisfy the requirements of Giesing’s Method of Images [Giesing, Kalman, and Rodden (1972a, Section 2.5.8)]. This example has a tapered fuselage, and is narrower forward in the region of the canard and wider aft at the root of the wing. The idealization selected chose the aft fuselage cross section as the reference for the interference tube which, in this example, is the circular cross section of the aft fuselage slender body. This idealization places its emphasis on modeling the aft fuselage interference effects, rather than on the effects of the canard. (In order to emphasize the effects of the canard, the interference tube would have chosen the fuselage cross section in the canard region as its reference.) With the larger diameter interference tube, the physical model has the canard root inside the interference tube. If the physical model were used for the data input, numerical singularities could arise because the external singularities of the lifting surface could be inside with the images. This situation violates the requirements of the Method of Images, which permits only a system of images inside the interference tube (and the line of axial doublets along the body

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centerline). The physical model of the lifting surfaces must be modified to maintain all parts of lifting surfaces outside of all interference tubes. A further judgment is required regarding which aeroelastic characteristics to emphasize: longitudinal or lateral-directional. This example emphasizes the longitudinal characteristics; therefore, the results will be more comparable to Example HA144A (and HA144E). The canard is raised vertically until its root intersects the interference tube at a height of 1.5 ft (recall that the fuselage has a radius of 2.0 ft at the root of the canard and the interference tube has a diameter of 2.5 ft). A new coordinate system, CORD2R 900, is introduced and the canard geometry is specified in this system. This coordinate system has its origin at the fuselage centerline, the extension of the canard leading edge, and an elevation of 1.5 ft relative to the basic coordinate system. This coordinate system moves the canard physically so that no part of it is within the interference tube and satisfies the requirements of the Method of Images. However, there is now a structural discontinuity that can be resolved by the use of multipoint constraints (MPCs). Four GRIDs are introduced at the intersections of the canard elastic axis and the sides of the fuselage; these are GRIDs 91 and 910 on the right side and GRIDs 89 and 890 on the left side. The right canard elastic axis CBAR 91 extends from the fuselage centerline GRID 90 to the side of the fuselage GRID 910 which is located in the basic coordinate system; the elastic axis continues outboard in CBAR 92 from GRIDs 91 to 92 located in CORD2R 900. GRIDs 91 and 910 are in identical locations but in different coordinate systems. The continuity of structural loads is assured by MPC 10, which provides the identity of all six components of GRIDS 91 and 910. Similarly, the left elastic axis, CBAR 90, connects GRIDs 90 and 890 within the fuselage in the basic coordinates, and CBAR 89 connects GRIDs 88 and 89 on the canard in CORD2R 900. Again, GRIDs 89 and 890 are identically located but in different coordinate systems, and MPC 10 also guarantees structural continuity by connecting all six components of GRIDs 89 and 890.

Trim Condition Only the level flight condition at sea level (Mach number m = 0.9 and dynamic pressure = 1200 psf) is considered in the present example (corresponding to Subcase 1 of previous Example HA144E) since this example is primarily intended to illustrate the inclusion of slender bodies in a quasi-static aeroelastic analysis. The effects of the pylons, bodies, and dihedral on the stability derivatives may be of interest, and a limited comparison with the longitudinal values of Example HA144E is shown in Table 6-5. The trim angle of attack and canard rotation are different from HA144E, not only because of the bodies but also because of the addition of incidence and dihedral. The aerodynamic and structural loads are not discussed here but are shown in Listing 6-25 as information for the user.

Case Control Commands The Case Control commands and the Executive Control statements are the same as in ExampleHA144E with the exception of the titles. The input data file echo for this example is shown below in Listing 6-23 followed by the sorted Bulk Data entries in Listing 6-24 and the output in Listing 6-25.

Output The significant output data in this example relate to the addition of the slender bodies to the configuration. The stability derivatives are compared to those from Example HA144E (without the bodies) in Table 6-5, and the aerodynamic loads on the bodies which are contained at the end of the AEROSTATIC DATA RECOVERY OUTPUT TABLE in Listing 6-25. The most significant differences relate to the canard, as would be expected.

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Aerodynamic Forces and Pressures Pressures are not available for the body elements but tabulated in place of the pressures are the values of μI , the strengths of the acceleration potential interference doublets, and μs elements are listed under NORMAL FORCES and MOMENTS and their components are labeled as YSB and ZSB for y-forces and z-forces, respectively. Their moments are zero because the grid point for all aerodynamic elements is at midchord and the distributed loading on a slender element has been assumed to act at this point. The eight fuselage element numbers are 4000 through 4007 and their side forces, YSB, are computed zeros because of the symmetry of the configuration and the maneuver; the vertical forces, ZSB, are all positive, upward in the aerodynamic coordinate system. The external store numbers are 4510 through 4513 on the right side, and 4610 through 4613 on the left. The side forces on both stores act away from the fuselage (the net pylon forces, LS 3500 and 3501 on the right and 3601 and 3601 on the left, also act away from the fuselage). The vertical store forces act upward on the forward three elements of each store but downward on the aft boattail elements as expected from Slender Body Theory.

Dihedral Effects As previously noted, the Vortex-Lattice method (VLM) in NX Nastran cannot predict all of the lateral-directional characteristics of wings. In this example, outboard dihedral was added to the wings. The dihedral effect refers to the rolling moment coefficient due to sideslip. The total dihedral effect of a wing should include the effects of geometric dihedral and its change under symmetric load factor and the effect of sweep. The first two effects can be found in NX Nastran. The geometric dihedral is accounted for in this example. The effect of load factor can be found by a restart that adds the previously determined deflections to the planform description on the wing CAERO1 entries, and the total geometric dihedral effect at the trim load factor will be obtained. The effect of sweep on the dihedral effect is a second order effect, depending on the lift coefficient, and can be estimated by the methods of Finck and Hoak (1976). In the case of general configurations with vertical surfaces, e.g., the fin and pylons in this example, NX Nastran also predicts their additional contributions to the dihedral effect. Table 6-5. Longitudinal Derivatives for Two Example FSW Airplanes Example HA144E Example HA144F Unrestrained Restrained Unrestrained Restrained Derevative Rigid Value at = Value at = Rigid Value Value at = Value at = Value 1200 psf 1200 psf 1200 psf 1200 psf 0.0 0.0 0.0 -0.007502 -0.009902 -0.012222 Czo 0.0 0.0 0.0 -0.004899 -0.006365 -0.007934 Cmo -5.071 -6.463 -7.788 -5.040 -6.537 -7.993 Czα -2.871 -3.667 -4.590 -2.878 -3.791 -4.777 Cmα -0.2461 -0.5430 -0.5181 -0.10379 -0.1600 -0.1445 Czδe 0.5715 0.3860 0.3980 0.2340 0.1988 0.2089 Cmδe Czq -12.074 -12.856 -16.166 -11.984 -13.248 -16.837 Cmq -9.954 -10.274 -12.545 -9.633 -10.352 -12.765 -

0.003809

-

-

0.004104

-

-

0.002738

-

-

0.002821

-

-

0.01521

-

-

0.016112

-

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Table 6-5. Longitudinal Derivatives for Two Example FSW Airplanes Example HA144E Example HA144F Unrestrained Restrained Unrestrained Restrained Derevative Rigid Value at = Value at = Rigid Value Value at = Value at = Value 1200 psf 1200 psf 1200 psf 1200 psf

αmo αm α α m δe αmq

-

0.009872

-

-

0.010478

-

-

0.0 0.9269 0.006723 -0.2121

-

-

-0.0001219 0.9258 0.002294 -0.2095

-

-

-0.0001225

-

-

-0.0001234

-

-

-0.001491

-

-

-0.001489

-

N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA144F $$$$$$ USERS GUIDE FOR AEROELASTIC ANALYSIS EXAMPLE HA144F $$$$$$ $ $ $ MODEL DESCRIPTION FULL SPAN 30 DEG FWD SWEPT WING $ $ WITH AILERON, CANARD AND AFT SWEPT $ $ VERTICAL FIN AND RUDDER. $ $ BAR MODEL WITH DUMBBELL MASSES. $ $ $ $ SOLUTION QUASI-STEADY AEROELASTIC ANALYSIS $ $ USING DOUBLET-LATTICE METHOD $ $ AERODYNAMICS AT MACH NO. 0.9. $ $ $ $ OUTPUT PLOTS OF THE STICK MODEL AND AERO $ $ GRID, LISTS OF RESTRAINED AND $ $ UNRESTRAINED SYMMETRIC AND ANTI$ $ SYMMETRIC STABILITY DERIVATIVES, $ $ AERODYNAMIC FORCES AND PRESSURES $ $ PLUS STRESSES AND DEFLECTIONS FOR $ $ LEVEL FLIGHT AND SEVERAL UNSYM$ $ METRICAL MANUEVERS. $ $ $ $ $ $$$$$$$$ $$$$$$$$ TIME 10 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO CEND EXAMPLE HA144F: FSW WITH FUSELAGE, 3 CONTROLS & 2 STORES UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL WITH DISPLACED CANARD C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144F: FSW WITH FUSELAGE, 3 CONTROLS & 2 STORES 2 SUBTI = UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO 3 LABEL = FULL-SPAN MODEL WITH DISPLACED CANARD 4 ECHO = BOTH 5 SPC = 1 $ SYMMETRIC CONSTRAINTS 6 MPC = 10 $ CANARD/FUSELAGE STRUCTURAL CONNECTIONS 7 DISP = ALL $ PRINT ALL DISPLACEMENTS 8 STRESS = ALL $ PRINT ALL STRESSES

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9 FORCE = ALL $ PRINT ALL FORCES 10 AEROF = ALL $ PRINT ALL AERODYNAMIC FORCES 11 APRES = ALL $ PRINT ALL AERODYNAMIC PRESSURES 12 SUBCASE 1 13 TRIM = 1 $ HIGH SPEED LEVEL FLIGHT 14 OUTPUT(PLOT) 15 PLOTTER = NASTRAN 16 SET 1 = ALL 17 FIND SCALE, ORIGIN 1,SET 1 18 PLOT SET 1 19 PLOT STATIC DEFORMATION 0, ORIGIN 1, SET 1, OUTLINE 20 BEGIN BULK EXAMPLE HA144F: FSW WITH FUSELAGE, 3 CONTROLS & 2 STORES UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL WITH DISPLACED CANARD I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ * * * STRUCTURAL DATA * * * $ $ $ $ (LB-FT-SEC SYSTEM) $ $ $ $ * * GRID GEOMETRY * * $ $ $ $ GRID 90 - 100 (T3) FUSELAGE POINTS $ $ GRID 110 - 122 (T3) WING POINTS $ $ GRID 310 - 312 (T3) FIN POINTS $ $ $ $ * FUSELAGE GRID * $ $ $ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $ $ ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS AND $ $ ITS ASSOCIATED SUPERELEMENT ID. $ $ $ $ ID CP X1 X2 X3 CD PS SEID GRID 90 15. 0. 0. GRID 97 0. 0. 0. GRID 98 10. 0. 0. GRID 99 20. 0. 0. GRID 100 30. 0. 0. GRID 890 15. -2.0 0. GRID 910 15. +2.0 0. $ $ $ * LEFT CANARD GRID * $ ID CP X1 X2 X3 CD PS SEID GRID 88 900 5.0 -5.0 0.0 GRID 89 900 5.0 -2.0 0.0 $ $ * RIGHT CANARD GRID * $ GRID 91 900 5.0 2.0 0.0 GRID 92 900 5.0 5.0 0.0 $ $ ** CANARD COORDINATE SYSTEMS ** $

10

.

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$ * CANARD HINGELINE AND ELASTIC AXIS * $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 90 900 5.0 0.0 0.0 5.0 0.0 15.0 $ C1 C2 C3 +CRD90C 20.0 0.0 0.0 $ $ * COORDINATES AT CANARD/INTERFERENCE TUBE INTERSECTION * $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 900 10.0 0.0 1.5 10.0 0.0 15.0 $ C1 C2 C3 +CRD900C20.0 0.0 1.5 $ $ * RIGHT WING GRID * $ $ ID CP X1 X2 X3 CD PS SEID GRID 111 24.61325 +5. 0. GRID 110 27.11325 +5. 0. GRID 112 29.61325 +5. 0. GRID 115 24.22650+10. 0. GRID 121 18.83975+15. 1. GRID 120 21.33975+15. 1. GRID 122 23.83975+15. 1. $ $ * RIGHT STORE GRID * $ GRID 150 19.22650+10. -1.5 GRID 151 24.22650+10. -1.5 $ $ * LEFT WING GRID * $ $ ID CP X1 X2 X3 CD PS SEID GRID 211 24.61325 -5. 0. GRID 210 27.11325 -5. 0. GRID 212 29.61325 -5. 0. GRID 215 24.22650-10. 0. GRID 221 18.83975-15. 1. GRID 220 21.33975-15. 1. GRID 222 23.83975-15. 1. $ $ * LEFT STORE GRID * $ GRID 250 19.22650-10. -1.5 GRID 251 24.22650-10. -1.5 $ $ * FIN GRID * $ GRID 310 32.88675+0. 5. GRID 311 30.38675+0. 5. GRID 312 35.38675+0. 5. $ $ * * STRUCTURAL STIFFNESS PROPERTIES * * $ $ * FUSELAGE STRUCTURE * $ $ THE CBAR ENTRY DEFINES A SIMPLE BEAM ELEMENT. LISTED ARE $ ITS PROPERTY ENTRY ID, THE TWO GRID POINTS JOINED BY THE $ BEAM AND COMPONENTS OF A VECTOR FROM THE FIRST POINT. $ THIS VECTOR DEFINES THE DIRECTION OF THE STRUCTURAL DE$ FLECTION OF THE POINT AND ITS POSITIVE SENSE. $ $ EID PID GA GB X1,GO X2 X3 CBAR 101 100 97 98 0. 0. 1. CBAR 102 100 98 90 0. 0. 1.

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+CRD +CRD90C

+CRD +CRD900C

$ $

$

$ $

$

$ $

$ $ $ $ $ $ $ $ $ $ $

Static Aeroelastic Analysis Sample Problems

CBAR CBAR $ $ $ $ $ $ $ $ $ $ $ $ $ PBAR $ +PB1F $ +PB2F $ $ $ CBAR CBAR CBAR CBAR $ $ $ MPC MPC MPC MPC MPC MPC $ $ $ MPC MPC MPC MPC MPC MPC $ $ $ $ CBAR CBAR CBAR $ $ $ $ $ $ $ $ RBAR RBAR RBAR RBAR $

100 103

100 100

90 99

99 100

0. 0.

0. 0.

1. 1.

THE PBAR ENTRY DEFINES GEOMETRIC PROPERTIES OF THE BEAM. LISTED ARE ITS ASSOCIATED MATERIAL ENTRY ID, ITS CROSS SECTIONAL AREA, AREA MOMENTS OF INERTIA, TORSIONAL MOMENT OF INERTIA AND NON-STRUCTURAL MASS PER UNIT AREA. THE OPTIONAL CONTINUATION ENTRY CONTAINS STRESS RECOVERY COEFFICIENTS, I.E., Y,Z COORDINATES WHERE STRESSES ARE TO BE COMPUTED. K1 AND K2 ARE AREA FACTORS FOR SHEAR STIFFNESS (DEFAULT IS BLANK; THEN SHEAR STIFFNESS IS INFINITE, I.E., SHEAR FLEXIBILITY IS ZERO. I12 IS THE AREA PRODUCT OF INERTIA. PID 100 C1 1.0 K1

MID 1 C2 1.0 K2

A 4.0 D1 1.0 I12 0.0

I1 .347222 D2 -1.0

I2 .30 E1 -1.0

J 1.0 E2 1.0

$ $ $ $ $ $ $ $ $ $ $ $

NSM +PB1F F1 -1.0

F2 -1.0

+PB2F

$ EID 89 90 91 92

PID 101 101 101 101

* CANARD STRUCTURE * GA GB X1,G0 88 89 0. 890 90 0. 90 910 0. 91 92 0.

X2 0. 0. 0. 0.

X3 1. 1. 1. 1.

* LEFT CANARD/FUSELAGE STRUCTURAL CONNECTION * 10 10 10 10 10 10

89 89 89 89 89 89

1 2 3 4 5 6

1.0 1.0 1.0 1.0 1.0 1.0

890 890 890 890 890 890

1 2 3 4 5 6

-1.0 -1.0 -1.0 -1.0 -1.0 -1.0

* RIGHT CANARD/FUSELAGE STRUCTURAL CONNECTION * 10 10 10 10 10 10

91 91 91 91 91 91

1 2 3 4 5 6

1.0 1.0 1.0 1.0 1.0 1.0

910 910 910 910 910 910

1 2 3 4 5 6

-1.0 -1.0 -1.0 -1.0 -1.0 -1.0

* RIGHT WING STRUCTURE * EID 110 115 120

PID 101 101 101

GA 100 110 115

GB 110 115 120

X1,GO 0. 0. 0.

X2 0. 0. 0.

$ $ X3 1. 1. 1.

THE RBAR ENTRY DEFINES A RIGID BAR. LISTED ARE THE GRID POINTS AT EACH END AND THE DEPENDENT AND INDEPENDENT DOFS AT EACH END. THE NUMBER OF INDEPENDENT DOFS AT THE TWO ENDS MUST EQUAL SIX. BY DEFAULT THOSE NOT DECLARED INDEPENDENT ARE MADE DEPENDENT. EID 111 112 121 122

GA 110 110 120 120

GB 111 112 121 122

CNA 123456 123456 123456 123456

CNB

CMA

$ $ $ $ $ $ $

CMB

$

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$ PBAR $ +PB1W $ +PB2W $ $ $ $ CBAR CBAR CBAR $ RBAR RBAR RBAR RBAR $ $ $ $ CBAR CBAR $ $ $ $ CBAR CBAR $ $ $ CBAR $ RBAR RBAR $ $ $ $ $ $ $ MAT1 $ $ $ $ $ $ $ $ $ $ $ CONM2 CONM2 CONM2 CONM2 $ $ $ CONM2 CONM2

6-112

PID 101 C1 0.5 K1

Static Aeroelastic Analysis Sample Problems

MID 1 C2 3.0 K2

A 1.5 D1 0.5 I12 0.0

I1 I2 0.173611+2.0 D2 E1 -3.0 -0.5

J NSM 0.462963 E2 F1 3.0 -0.5

+PB1W F2 -3.0

$ $ $

* LEFT WING STRUCTURE * EID 210 215 220

PID 101 101 101

GA 100 210 215

GB 210 215 220

211 212 221 222

210 210 220 220

211 212 221 222

123456 123456 123456 123456

X1,GO 0. 0. 0.

+PB2W

X2 0. 0. 0.

X3 1. 1. 1.

$ * RIGHT

PYLON/STORE STRUCTURE *

$ $

EID 150 151

PID 101 101

GA 150 151

GB 151 115

X1,GO 0. 1.

X2 0. 0.

X3 1. 0.

* LEFT PYLON/STORE STRUCTURE * EID 250 251

PID 101 101

GA 250 251

GB 251 215

X1,GO 0. 1.

X2 0. 0.

$ $ X3 1. 0.

* FIN STRUCTURE * 310

101

100

310

311 312

310 310

311 312

123456 123456

0.

$ $ 0.

1. $

THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES. LISTED ARE ITS ID, ITS ELASTIC MODULUS, SHEAR MODULUS, POISSONS RATIO, MASS DENSITY, TEMPERATURE EXPANSION COEFFICIENT, REFERENCE TEMPERATURE AND A STRUCTURAL DAMPING COEFFICIENT. MID 1

E 1.44+9

G 5.40+8

NU

RHO

A

TREF

GE

* * MASS AND INERTIA PROPERTIES * * * FUSELAGE MASSES * THE CONM2 ENTRY DEFINES A CONCENTRATED MASS. LISTED ARE ITS ID, GRID LOCATION, COORDINATE SYSTEM TO LOCATE THE CENTER OF GRAVITY, THE MASS VALUE AND THE LOCATION OF THE CENTER OF GRAVITY RELATIVE TO THE GRID LOCATION. EID 97 98 99 100

G 97 98 99 100

CID 0 0 0 0

M 3000.0 3000.0 3000.0 3000.0

X1

* RIGHT WING MASSES * 111 112

111 112

0 0

600.0 400.0

NX Nastran Aeroelastic Analysis User’s Guide

X2

$ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $

X3

$ $ $

Static Aeroelastic Analysis Sample Problems

CONM2 CONM2 $ $ $ CONM2 CONM2 CONM2 CONM2 $ $ $ CONM2 CONM2 $ $ $ $ $ $ $ $ $ PARAM PARAM $ $ $ $ $ $ $ SPC1 $ $ $ $ $ $ $ $ $ $ SUPORT $ $ $ $ $ OMIT1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

121 122

121 122

0 0

600.0 400.0 $ $ $

* LEFT WING MASSES * 211 212 221 222

211 212 221 222

0 0 0 0

600.0 400.0 600.0 400.0 $ $ $

* FIN MASSES * 311 312

311 312

0 0

60.0 40.0

* * STRUCTURAL PARAMETERS * * THE PARAM,WTMASS,GINV CAUSES ALL THE STRUCTURAL MASSES AND MASS DENSITIES TO BE MULTIPLIED BY GINV (I.E., BY ONE OVER THE ACCELERATION OF GRAVITY). THE DYNAMIC PRESSURE SUPPLIED FOR AERODYNAMIC FORCE CALCULATIONS WILL NOT BE MULTIPLIED BY GINV. WTMASS AUNITS

.031081 .031081 * * STRUCTURAL CONSTRAINTS * *

THE SPC1 ENTRY CONSTRAINS THE LISTED GRID POINTS IN THE SPECIFIED DOF COMPONENTS. SID 1

C 1

G1 90

G2

G3

G 110

G 120

G 210

G 220

(LB-FT-SEC SYSTEM) * * ELEMENT GEOMETRY * * THE AEROS ENTRY IS UNIQUE TO THE STATIC AEROELASTICITY SOLUTION, SOL21. ACSID IDENTIFIES THE AERO COORDINATE SYSTEM, RCSID IDENTIFIES THE REFERENCE COORDINATE SYSTEM FOR RIGID BODY MOTION. REFC IS THE REFERENCE CHORD. REFB IS THE REFERENCE SPAN. REFS IS THE REFERENCE WING AREA. SYMXZ AND SYMXY ARE SYMMETRY KEYS. RCSID

REFC

REFB

REFS

$ $ $ $

310

* * * AERODYNAMIC DATA * * *

ACSID

$ $ $ $ $ $ $ $ $

C 23456

THE OMIT1 ENTRY IDENTIFIES GRID POINTS TO BE OMITTED FROM THE REMAINDER OF THE ANALYSIS. ID 4

$ $ $ $ $ $

G4

THE SUPORT ENTRY IDENTIFIES A GRID POINT OR A SCALAR POINT AND SPECIFIES THE DOF COMPONENTS IN WHICH THE USER DESIRES REACTIONS TO BE APPLIED TO PREVENT RIGID BODY MOTION. IT THUS INVOKES THE SOLUTION OF THE BALANCE EQUATIONS TO DETERMINE THE REACTIONS. IN THE STATIC AEROELASTIC SOLUTION THE DOF COMPONENTS MUST BE CONSISTENT WITH THE UNDEFINED VARIABLES ON THE TRIM ENTRIES. ID 90

$ $ $ $ $ $ $ $ $

SYMXZ

$ $ $ $ $ $ $ $ $ $ $ $ $ $

SYMXY

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Chapter 6

Static Aeroelastic Analysis Sample Problems

AEROS 1 100 10.0 40.0 400.0 $ $ THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM $ FLAGGED BY THE AEROS ENTRY. THE ORIGIN IS AT THE CANARD $ QUARTER CHORD. LISTED ARE THE ORIGIN, A POINT ALONG THE $ Z AXIS AND A POINT IN THE X-Z PLANE, ALL IN THE RID $ COORDINATE SYSTEM. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0 12.5 0. 0. 12.5 0. 10. $ C1 C2 C3 +CRD1 20. 0. 0. $ $ THIS CORD2R ENTRY DEFINES THE NACA COORDINATE SYSTEM TO $ WHICH ALL THE STABILITY DERIVATIVES AND TRIM CONDITIONS $ WILL BE REFERENCED. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 -10.0 $ C1 C2 C3 +CRD100 0.0 0.0 0.0 $ $ $ * WING AERODYNAMIC MODEL * $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AERODYNAMICS. $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS, $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ EXTRA POINT IDS. $ $ * RIGHT INBOARD WING * $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 1104 1000 3 4 1 $ ( FWD LEFT POINT ) CHORD ( FWD RIGHT POINT ) CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CARW2 23.55662+2.5 0. 10. 19.22650+10. 0. 10. $ $ * RIGHT OUTBOARD WING * $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 1116 1000 4 4 1 $ ( FWD LEFT POINT ) CHORD ( FWD RIGHT POINT ) CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CARW3 19.22650+10. 0. 10. 13.45299+20. 2. 10. $ $ $ * LEFT OUTBOARD WING * CAERO1 2100 1000 4 4 1 +CALW1 13.45299-20. 2. 10. 19.22650-10. 0. 10. $ $ * LEFT INBOARD WING * CAERO1 2116 1000 3 4 1 +CALW2 19.22650-10. 0. 10. 23.55662-2.5 0. 10. $ $ $ * CANARD AERODYNAMIC MODEL * $

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NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ +CRD1

$ $ $ $ $ +CRD100

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

+CARW2

$

+CARW3

$ $ +CALW1 $ +CALW2 $ $ $

Static Aeroelastic Analysis Sample Problems

$ CAERO1 +CARC2 $ AEFACT $ $ CAERO1 +CALC1 $ AEFACT $ $ $ $ CAERO1 +CA1FI1 $ $ $ $ $ $ PAERO1 $ $ CAERO2 $ +CA2F $ $ PAERO2 $ +PA2F $ $ $ AEFACT +LRSBF $ $ $ AEFACT $ $ $ $ $ CAERO1 +CA1RP $ $ $ $ CAERO2 $ +CA2RS $ $ PAERO2 $ +PA2RS $ $ $

* RIGHT OUTBOARD SIDE * 4 1000 10.0 0.0 5.0

1000 0.0

1000 2.0

900 0.0

1000

0.0

0.16667 1.0

2000 0.0

1000 -5.0

900 0.0

2000

0.0

0.83333 1.0

0.0

1 10.

+CARC2 $

* LEFT OUTBOARD SIDE * 4 2000 10. 0.0 -2.0

0.0

1 10.

+CALC1 $

* FIN AERODYNAMIC MODEL * 3100 1000 30.7735 0.

10.

3 10.

4 26.44334 0.0

2.5

1 10.

$ $ $ +CA1FI1 $

* FUSELAGE AERODYNAMIC MODEL * PID 1000

B1 4000

B2 4510

B3 4610

B4

B5

B6

EID 4000 X1 -5.0

PID 4020 Y1 0.0

CP

NINT 8

LSB

LINT

IGID 1

+CA2 +CA2F

Z1 0.0

NSB 8 X12 40.0

PID 4020 THI1 1

ORIENT ZY THN1 8

WIDTH 2.5 THI2

AR 1.0 THN2

LRSB 4015

LRIB

LTH1 4018

LTH2

$ +PA2 +PA2F

2.5

+LRSBF

1 5.0

+CA1RP

* (LRSB) SLENDER BODY HALF-WIDTHS * 4015 2.5

0.0 2.5

1.111

1.778

2.0

2.0

2.0

* (LTH1 & 2) THETA ARRAYS * 4018

45.

135.

225.

315.

* RIGHT PYLON/STORE AERODYNAMIC MODEL *

3500 1000 21.72650+10.0

0.

* RIGHT PYLON * 1 2 5.0 21.72650+10.0

-1.0

$ * RIGHT STORE * EID PID 4510 4520 X1 Y1 19.22650+10.0 PID 4520 THI1 1

ORIENT ZY THN1 4

CP

NINT 4

LSB

LINT

IGID 1

+CA2 +CA2RS

Z1 -1.5

NSB 4 X12 10.0

WIDTH 0.5 THI2

AR 1.0 THN2

LRSB 4515

LRIB

LTH1 4018

LTH2

$ +PA2 +PA2RS

* (LRSB) SLENDER BODY HALF-WIDTHS *

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Chapter 6

AEFACT $ $ $ $ $ CAERO1 +CA1LP $ $ $ $ CAERO2 $ +CA2LS $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ SPLINE2 $ +SPRW2 $ $ $ SPLINE2 $ +SPRW3 $ $ $ $ $ SET1 SET1 $ $ $ SPLINE2 $ +SPLW1 $ $ $ SPLINE2 $ +SPLW2 $ $ $ $

6-116

4515

Static Aeroelastic Analysis Sample Problems

0.0

0.5

0.5

0.5

0.0

* LEFT PYLON/STORE AERODYNAMIC MODEL *

3600 1000 21.72650-10.0

0.

* LEFT PYLON * 1 2 5.0 21.72650-10.0

-1.0

1 5.0

+CA1LP $

* LEFT STORE * EID PID 4610 4520 X1 Y1 19.22650-10.0

CP Z1 -1.5

NSB 4 X12 10.0

NINT 4

LSB

LINT

IGID 1

+CA2 +CA2LS

$

* * SPLINE FIT ON THE LIFTING SURFACES * * * BEAM SPLINE FIT ON THE WING * THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLATION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR ATTACHMENT AND TORSIONAL FLEXIBILITIES. CID IDENTIFIES THE CORD2R ENTRY THAT DEFINES THE SPLINE AXIS. DTHX AND DTHY ARE ROTATIONAL ATTACHMENT FLEXIBILITIES (-1. SPECIFIES NO ATTACHMENT).

EID 1602 DTHX -1.

EID 1603 DTHX -1.

CAERO 1104 DTHY -1.

CAERO 1116 DTHY -1.

ID1 1104

* RIGHT INBOARD WING * ID2 SETG DZ 1115 1101 0.

* RIGHT OUTBOARD WING * ID1 ID2 SETG DZ 1116 1131 1102 0.

DTOR 1.

CID 2

EID 2601 DTHX -1.

EID 2602 DTHX -1.

G1 100 115

CAERO 2100 DTHY -1.

CAERO 2116 DTHY -1.

G2 111 121

G3 112 122

DTOR 1.

CID 3

+SPRW3

$ $ $ $

G4 115

* LEFT OUTBOARD WING * ID1 ID2 SETG DZ 2100 2115 2102 0.

* LEFT INBOARD WING * ID1 ID2 SETG DZ 2116 2127 2101 0.

$ $ DTOR 1.

CID 30

+SPLW1

$ $ DTOR 1.

CID 20

THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION.

NX Nastran Aeroelastic Analysis User’s Guide

+SPRW2

$ $

THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. SID 1101 1102

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

+SPLW2

$ $ $ $

Static Aeroelastic Analysis Sample Problems

$ SET1 SET1 $ $ $ $ $ $ SPLINE2 +SPRC2 $ SET1 $ $ SPLINE2 +SPLC1 $ $ $ SPLINE2 +SP2FI1 $ SET1 $ $ $ $ $ SPLINE2 +SP2RP $ SET1 $ $ $ SPLINE2 +SP2LP $ SET1 $ $ $ $ $ $ SPLINE2 +SP2F $ SET1 $ $ $ $ SPLINE2 +SP2RS $ SET1 $ $ $ $ SPLINE2 +SP2LS $ SET1

SID 2101 2102

G1 100 215

G2 211 221

G3 212 222

G4 215 $

* BEAM SPLINE FIT ON THE CANARD *

1502 0.0

1000 0.0

1000

88

* RIGHT OUTBOARD SIDE * 1000 1007 1000 0.

1.

90

$ $ $ $ +SPRC2 $

2501 0.0

2000 0.0

89

91

92

* LEFT OUTBOARD SIDE * 2000 2007 1000

0.

1.

90

$ $ +SPLC1

300

$ $ $ +SP2FI1

* BEAM SPLINE FIT ON THE FIN * 3100 -1.

3100 -1.

3100

3111

3100

3100

99

100

311

312

0.

1.

$ $ $ $

* BEAM SPLINE FIT ON THE PYLONS * * RIGHT PYLON * 4520 -1.

3500 -1.

3500

3501

4521

115

150

151

4521

0.

1.

450

+SP2RP $ $

* LEFT PYLON * 4620 -1.

3600 -1.

3600

3601

4621

215

250

251

4621

0.

1.

460

+SP2LP $ $

* BEAM SPLINE FIT ON THE BODIES *

$ $

* FUSELAGE * 4000

4000 -1.

4000

4007

4001

4001

97

98

99

100

0.

1.

+SP2F $ $ $

* RIGHT STORE * 4525

4510 -1.

4510

4525

150

151

4513

4525

0.

1.

+SP2RS $ $ $

* LEFT STORE * 4625

4610 -1.

4610

4625

250

251

4613

4625

0.

1.

+SP2LS $

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 6

Static Aeroelastic Analysis Sample Problems

$ $ $ THE CORD2R ENTRY DEFINES THE COORDINATE SYSTEM IN WHICH THE $ $ BEAM SPLINE EXTENDS ALONG THE WING Y-AXIS. IT LISTS THE $ $ ORIGIN, A POINT ALONG THE Z-AXIS AND A POINT IN THE X-Z $ $ PLANE. $ $ $ $ * INBOARD RIGHT WING SPLINE AXIS * $ $ $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 2 0 30. 0. 0. 30. 0. 10. +CRD2RW $ C1 C2 C3 +CRD2RW 38.66025+5.0 0. $ $ $ * OUTBOARD RIGHT WING SPLINE AXIS * $ $ $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 3 0 24.22650+10. 0. 25.07982 8.293369.853293+CRD2ORW $ C1 C2 C3 +CRD2ORW34.2265015.77350+0. $ $ $ * INBOARD LEFT WING SPLINE AXIS * $ $ $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 20 0 30.0 0. 0. 30. 0. 10. +CRD2ILW $ C1 C2 C3 +CRD2ILW38.66025-5.0 0. $ $ $ * OUTBOARD LEFT WING SPLINE AXIS * $ $ $ $ CID CS A1 A2 A3 B1 B2 B3 CORD2R 30 0 24.22650-10. 0. 25.07982-8.293369.853293+CRD2OLW $ C1 C2 C3 +CRD2OLW34.22650-15.7735+0. $ $ $ * FIN SPLINE AXIS * $ $ $ CORD2R 300 0 30.0 0. 0. 30.0 10.0 0. +CRD2FI +CRD2FI 20.0 0.0 5.7735 $ $ * RIGHT PYLON SPLINE AXIS * $ CORD2R 450 0 24.22650+10.0 0. 24.22650+20.0 0.0 +CRD2RP +CRD2RP 30.0 20.0 0.0 $ $ * LEFT PYLON SPLINE AXIS * $ $ CORD2R 460 0 24.22650-10.0 0. 24.22650+20.0 0.0 +CRD2LP +CRD2LP 30.0 20.0 0.0 $ $ $ $ * CONTROL SURFACE DEFINITION * $ $ $ $ THE AESURF ENTRY DEFINES AN AERODYNAMIC CONTROL SURFACE. $ $ LISTED ARE THE ALPHANUMERIC NAME OF THE SURFACE, THE ID $ $ OF A COORDINATE SYSTEM THAT DEFINES THE HINGE LINE AND $ $ THE ID OF AN AELIST ENTRY. $ $ $ $ ID LABEL CID1 ALID1 CID2 ALID2 AESURF 505 ELEV 90 1000 90 2000 AESURF 517 AILERON 110 1100 210 2100 AESURF 518 RUDDER 301 3000 $ $ $ THE AELIST ENTRY LISTS AERODYNAMIC BOXES THAT LIE ON THE $ $ CONTROL SURFACE. $ $ $

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Static Aeroelastic Analysis Sample Problems

$ AELIST AELIST AELIST AELIST AELIST $ $ $ $ $ $ $ $ CORD2R +CRD2RA $ $ CORD2R +CRD2LA $ $ CORD2R +CRD2R $ $ $ $ $ $ $ $ $ $ $ $ $ $ * $ $ DMI $ DMI +DM1 +DM2 $ $ $ $ $ $ $ $ $ $ $ $ AESTAT AESTAT AESTAT AESTAT AESTAT AESTAT AESTAT AESTAT

SID 1000 2000 1100 2100 3000

E1 1000 2000 1119 2103 3103

E2 THRU THRU 1123 2107 3107

E3 1007 2007 1127 2111 3111

ETC

1131 2115

$ $ $ * CANARD * $ THE COORDINATE SYSTEM, CORD2R,1, REFERENCED BY THE AEROS ENTRY $ IS THE CANARD HINGE LINE, AND NEEDS NO FURTHER DEFINITION $ $ * RIGHT AILERON * $ 110 0 26.7265 10.0 0. 26.7265 10.0 -10.0 +CRD2RA 36.7265 15.7735 0. $ * LEFT AILERON * $ 210 0 26.7265 -10.0 0. 26.7265 -10.0 10.0 +CRD2LA 36.7265 -15.7735 0. $ * RUDDER * $ 301 0 32.5 0. 0. 32.5 -10. 0.0 +CRD2R 22.5 0. 5.7735 $ * * USER SUPPLIED INPUT DATA * * $ $ THE DMI ENTRY ACCOMMODATES DIRECT INPUT OF USER SUPPLIED $ MATRICES OF DATA. LISTED ARE THE NAME OF THE MATRIX, THE $ FORM OF MATRIX (IN THIS CASE DIAGONAL), THE TYPE OF DATA $ (IN THIS CASE REAL SINGLE PRECISION), BEING INPUT AND THE $ TYPE EXPECTED AT OUTPUT (IN THIS CASE TO BE DETERMINED $ INTERNALLY). M IS THE NUMBER OF ROWS AND N IS THE NUMBER $ OF COLUMNS. THE DATA IS EXPECTED BY COLUMNS. THE CONTIN$ UATION ENTRY LISTS THE COLUMN NO., THE ROW NO. OF THE FIRST $ NON-ZERO ELEMENT AND THE FOLLOWING ELEMENTS IN THAT COLUMN. $ $ INITIAL DOWNWASHES (E.G., DUE TO INCIDENCE,TWIST OR CAMBER) * $ $ NAME "0" FORM TIN TOUT M N W2GJ 0 2 1 0 152 1 NAME J I1 A(I1,J) A(I1+1,J)... W2GJ 1 9 .0017453THRU 20 21 .0017114+DM1 THRU 36 45 .0017114THRU 60 61 .0017453+DM2 THRU 72 * CONTROL SURFACE HINGE LINES *

* * * SOLUTION SPECIFICATIONS * * * * * AERODYNAMIC DOFS * * THE AESTAT CARD LISTS TRIM VARIABLES USED TO SPECIFY RIGID BODY MOTIONS. THESE AND THE CONTROL SURFACE ROTATIONS MAKE UP THE VARIABLES IN THE EQUATIONS OF MOTION. ID 501 502 503 504 511 512 513 514

$ $ $ $ $ $ $ $ $ $

LABEL ANGLEA PITCH URDD3 URDD5 SIDES YAW ROLL URDD2

NX Nastran Aeroelastic Analysis User’s Guide

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Static Aeroelastic Analysis Sample Problems

AESTAT 515 URDD4 AESTAT 516 URDD6 $ $ * * TRIM CONDITIONS * * $ $ THE TRIM ENTRY SPECIFIES CONSTRAINTS FOR THE TRIM VARIABLES $ LISTED ON THE AESTAT AND AESURF ENTRIES. LISTED ARE ITS ID, $ THE MACH NUMBER, DYNAMIC PRESSURE AND PAIRS OF TRIM VARI$ ABLES AND THEIR CONSTRAINED VALUES. THOSE THAT ARE NOT $ HELD FIXED MUST BE CONSTRAINED BY REACTION FORCES STIPU$ LATED ON THE SUPORT ENTRY. SEE SECTION 3.5.3 OF THE THEO$ RETICAL MANUAL FOR MORE DETAILS. $ $ TRIM CONDITION 1: LEVEL FLIGHT AT HIGH DYNAMIC PRESSURE $ TRIM 1 0.9 1200.0 PITCH 0. URDD3 -1.0 +TR1A URDD5 0. AILERON 0. RUDDER 0. URDD2 0. +TR1B URDD4 0. URDD6 0. $ $ * * * ENDDATA INPUT BULK DATA CARD COUNT = 722

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NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ $ $ $ $ $ +TR1A +TR1B $ $

Static Aeroelastic Analysis Sample Problems

Listing 6-23. Input Files for FSW Airplane with Bodies EXAMPLE HA144F: FSW WITH FUSELAGE, 3 CONTROLS & 2 STORES PAGE 18 UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL WITH DISPLACED CANARD S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 1000 0.0 0.16667 1.0 2AEFACT 2000 0.0 0.83333 1.0 3AEFACT 4015 0.0 1.111 1.778 2.0 2.0 2.0 2.5 +LRSBF 4+LRSBF 2.5 2.5 5AEFACT 4018 45. 135. 225. 315. 6AEFACT 4515 0.0 0.5 0.5 0.5 0.0 7AELIST 1000 1000 THRU 1007 8AELIST 1100 1119 1123 1127 1131 9AELIST 2000 2000 THRU 2007 10AELIST 2100 2103 2107 2111 2115 11AELIST 3000 3103 3107 3111 12AEROS 1 100 10.0 40.0 400.0 13AESTAT 501 ANGLEA 14AESTAT 502 PITCH 15AESTAT 503 URDD3 16AESTAT 504 URDD5 17AESTAT 511 SIDES 18AESTAT 512 YAW 19AESTAT 513 ROLL 20AESTAT 514 URDD2 21AESTAT 515 URDD4 22AESTAT 516 URDD6 23AESURF 505 ELEV 90 1000 90 2000 24AESURF 517 AILERON 110 1100 210 2100 25AESURF 518 RUDDER 301 3000 26CAERO1 1000 1000 900 4 1000 1 +CARC2 27+CARC2 0.0 2.0 0.0 10.0 0.0 5.0 0.0 10. 28CAERO1 1104 1000 3 4 1 +CARW2 29+CARW2 23.55662+2.5 0. 10. 19.22650+10. 0. 10. 30CAERO1 1116 1000 4 4 1 +CARW3 31+CARW3 19.22650+10. 0. 10. 13.45299+20. 2. 10. 32CAERO1 2000 1000 900 4 2000 1 +CALC1 33+CALC1 0.0 -5.0 0.0 10. 0.0 -2.0 0.0 10. 34CAERO1 2100 1000 4 4 1 +CALW1 35+CALW1 13.45299-20. 2. 10. 19.22650-10. 0. 10. 36CAERO1 2116 1000 3 4 1 +CALW2 37+CALW2 19.22650-10. 0. 10. 23.55662-2.5 0. 10. 38CAERO1 3100 1000 3 4 1 +CA1FI1 39+CA1FI1 30.7735 0. 10. 10. 26.443340.0 2.5 10. 40CAERO1 3500 1000 1 2 1 +CA1RP 41+CA1RP 21.72650+10.0 0. 5.0 21.72650+10.0 -1.0 5.0 42CAERO1 3600 1000 1 2 1 +CA1LP 43+CA1LP 21.72650-10.0 0. 5.0 21.72650-10.0 -1.0 5.0 44CAERO2 4000 4020 8 8 1 +CA2F 45+CA2F -5.0 0.0 0.0 40.0 46CAERO2 4510 4520 4 4 1 +CA2RS 47+CA2RS 19.22650+10.0 -1.5 10.0 48CAERO2 4610 4520 4 4 1 +CA2LS 49+CA2LS 19.22650-10.0 -1.5 10.0 50CBAR 89 101 88 89 0. 0. 1. 51CBAR 90 101 890 90 0. 0. 1. 52CBAR 91 101 90 910 0. 0. 1. 53CBAR 92 101 91 92 0. 0. 1. 54CBAR 100 100 90 99 0. 0. 1. 55CBAR 101 100 97 98 0. 0. 1. 56CBAR 102 100 98 90 0. 0. 1. 57CBAR 103 100 99 100 0. 0. 1. 58CBAR 110 101 100 110 0. 0. 1. 59CBAR 115 101 110 115 0. 0. 1. 60CBAR 120 101 115 120 0. 0. 1. 61CBAR 150 101 150 151 0. 0. 1. 62CBAR 151 101 151 115 1. 0. 0. 63CBAR 210 101 100 210 0. 0. 1. 64CBAR 215 101 210 215 0. 0. 1. 65CBAR 220 101 215 220 0. 0. 1. 66CBAR 250 101 250 251 0. 0. 1. 67CBAR 251 101 251 215 1. 0. 0. 68CBAR 310 101 100 310 0. 0. 1. 69CONM2 97 97 0 3000.0 70CONM2 98 98 0 3000.0 71CONM2 99 99 0 3000.0 72CONM2 100 100 0 3000.0 73CONM2 111 111 0 600.0 74CONM2 112 112 0 400.0 75CONM2 121 121 0 600.0 76CONM2 122 122 0 400.0 77CONM2 211 211 0 600.0 78CONM2 212 212 0 400.0 79CONM2 221 221 0 600.0 80CONM2 222 222 0 400.0 81CONM2 311 311 0 60.0 82CONM2 312 312 0 40.0 83CORD2R 1 0 12.5 0. 0. 12.5 0. 10. +CRD1 84+CRD1 20. 0. 0. 85CORD2R 2 0 30. 0. 0. 30. 0. 10. +CRD2RW NX Nastran Aeroelastic Analysis User’s Guide 86+CRD2RW 38.66025+5.0 0. 87CORD2R 3 0 24.22650+10. 0. 25.079828.29336 9.853293+CRD2ORW 88+CRD2ORW34.2265015.77350+0.

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919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181-

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Static Aeroelastic Analysis Sample Problems

CORD2R 30 0 24.22650-10. 0. +CRD2OLW34.22650-15.7735+0. CORD2R 90 900 5.0 0.0 0.0 +CRD90C 20.0 0.0 0.0 CORD2R 100 0 15.0 0.0 0.0 +CRD100 0.0 0.0 0.0 CORD2R 110 0 26.7265 10.0 0. +CRD2RA 36.7265 15.7735 0. CORD2R 210 0 26.7265 -10.0 0. +CRD2LA 36.7265 -15.77350. CORD2R 300 0 30.0 0. 0. +CRD2FI 20.0 0.0 5.7735 CORD2R 301 0 32.5 0. 0. +CRD2R 22.5 0. 5.7735 CORD2R 450 0 24.22650+10.0 0. +CRD2RP 30.0 20.0 0.0 CORD2R 460 0 24.22650-10.0 0. +CRD2LP 30.0 20.0 0.0 CORD2R 900 10.0 0.0 1.5 +CRD900C20.0 0.0 1.5 DMI W2GJ 0 2 1 0 DMI W2GJ 1 9 .0017453THRU +DM1 THRU 36 45 .0017114THRU +DM2 THRU 72 GRID 88 900 5.0 -5.0 0.0 GRID 89 900 5.0 -2.0 0.0 GRID 90 15. 0. 0. GRID 91 900 5.0 2.0 0.0 GRID 92 900 5.0 5.0 0.0 GRID 97 0. 0. 0. GRID 98 10. 0. 0. GRID 99 20. 0. 0. GRID 100 30. 0. 0. GRID 110 27.11325+5. 0. GRID 111 24.61325+5. 0. GRID 112 29.61325+5. 0. GRID 115 24.22650+10. 0. GRID 120 21.33975+15. 1. GRID 121 18.83975+15. 1. GRID 122 23.83975+15. 1. GRID 150 19.22650+10. -1.5 GRID 151 24.22650+10. -1.5 GRID 210 27.11325-5. 0. GRID 211 24.61325-5. 0. GRID 212 29.61325-5. 0. GRID 215 24.22650-10. 0. GRID 220 21.33975-15. 1. GRID 221 18.83975-15. 1. GRID 222 23.83975-15. 1. GRID 250 19.22650-10. -1.5 GRID 251 24.22650-10. -1.5 GRID 310 32.88675+0. 5. GRID 311 30.38675+0. 5. GRID 312 35.38675+0. 5. GRID 890 15. -2.0 0. GRID 910 15. +2.0 0. MAT1 1 1.44+9 5.40+8 MPC 10 89 1 1.0 890 MPC 10 89 2 1.0 890 MPC 10 89 3 1.0 890 MPC 10 89 4 1.0 890 MPC 10 89 5 1.0 890 MPC 10 89 6 1.0 890 MPC 10 91 1 1.0 910 MPC 10 91 2 1.0 910 MPC 10 91 3 1.0 910 MPC 10 91 4 1.0 910 MPC 10 91 5 1.0 910 MPC 10 91 6 1.0 910 OMIT1 4 110 120 210 220 PAERO1 1000 4000 4510 4610 PAERO2 4020 ZY 2.5 1.0 4015 +PA2F 1 8 PAERO2 4520 ZY 0.5 1.0 4515 +PA2RS 1 4 PARAM AUNITS .031081 PARAM WTMASS .031081 PBAR 100 1 4.0 .347222 .30 +PB1F 1.0 1.0 1.0 -1.0 -1.0 +PB2F 0.0 PBAR 101 1 1.5 0.173611+2.0 +PB1W 0.5 3.0 0.5 -3.0 -0.5 +PB2W 0.0 RBAR 111 110 111 123456 RBAR 112 110 112 123456 RBAR 121 120 121 123456 RBAR 122 120 122 123456 RBAR 211 210 211 123456 RBAR 212 210 212 123456 RBAR 221 220 221 123456 RBAR 222 220 222 123456

NX Nastran Aeroelastic Analysis User’s Guide

25.07982-8.293369.853293+CRD2OLW 5.0

0.0

15.0

+CRD90C

15.0

0.0

-10.0

+CRD100

26.7265 10.0

-10.0

+CRD2RA

26.7265 -10.0

10.0

+CRD2LA

30.0

10.0

0.

+CRD2FI

32.5

-10.

0.0

+CRD2R

24.22650+20.0

0.0

+CRD2RP

24.22650+20.0

0.0

+CRD2LP

10.0

0.0

15.0

+CRD900C

20 60

152 21 61

1 .0017114+DM1 .0017453+DM2

1 2 3 4 5 6 1 2 3 4 5 6 310

-1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0

1.0 1.0

4018

+PA2F

4018

+PA2RS

-1.0

-1.0

+PB1F +PB2F

0.462963 3.0 -0.5

-3.0

+PB1W +PB2W

Static Aeroelastic Analysis Sample Problems

182RBAR 311 183RBAR 312 184SET1 1000 185SET1 1101 186SET1 1102 187SET1 2101 188SET1 2102 189SET1 3100 190SET1 4001 191SET1 4521 192SET1 4525 193SET1 4621 194SET1 4625 195SPC1 1 196SPLINE2 1502 197+SPRC2 0.0 198SPLINE2 1602 199+SPRW2 -1. 200SPLINE2 1603 201+SPRW3 -1. 202SPLINE2 2501 203+SPLC1 0.0 204SPLINE2 2601 205+SPLW1 -1. 206SPLINE2 2602 207+SPLW2 -1. 208SPLINE2 3100 209+SP2FI1 -1. 210SPLINE2 4000 211+SP2F 212SPLINE2 4520 213+SP2RP -1. 214SPLINE2 4525 215+SP2RS 216SPLINE2 4620 217+SP2LP -1. 218SPLINE2 4625 219+SP2LS 220SUPORT 90 221TRIM 1 222+TR1A URDD5 223+TR1B URDD4 ENDDATA TOTAL COUNT= 224

310 310 88 100 115 100 215 99 97 115 150 215 250 1 1000 0.0 1104 -1. 1116 -1. 2000 0.0 2100 -1. 2116 -1. 3100 -1. 4000 -1. 3500 -1. 4510 -1. 3600 -1. 4610 -1. 23456 0.9 0. 0.

311 312 89 111 121 211 221 100 98 150 151 250 251 90 1000

123456 123456 91 112 122 212 222 311 99 151

1007

1000

0.

1.

90

+SPRC2

1104

1115

1101

0.

1.

2

+SPRW2

1116

1131

1102

0.

1.

3

+SPRW3

2000

2007

1000

0.

1.

90

+SPLC1

2100

2115

2102

0.

1.

30

+SPLW1

2116

2127

2101

0.

1.

20

+SPLW2

3100

3111

3100

0.

1.

300

4000

4007

4001

0.

1.

3500

3501

4521

0.

1.

4510

4513

4525

0.

1.

3600

3601

4621

0.

1.

4610

4613

4625

0.

1.

0. RUDDER

URDD3 0.

-1.0 URDD2

92 115 215 312 100

251

1200.0 PITCH AILERON 0. URDD6 0.

+SP2FI1 +SP2F

450

+SP2RP +SP2RS

460

+SP2LP +SP2LS

0.

+TR1A +TR1B

NX Nastran Aeroelastic Analysis User’s Guide

6-123

Chapter 6

Static Aeroelastic Analysis Sample Problems

Listing 6-24. Sorted Bulk Data Entries for FSW Airplane with Bodies EXAMPLE HA144F: FSW WITH FUSELAGE, 3 CONTROLS & 2 STORES UNSYMMETRIC FLIGHT CONDITIONS, DOUBLET-LATTICE AERO FULL-SPAN MODEL WITH DISPLACED CANARD SUBCASE 1 N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 9.0000E-01 Q = 1.2000E+03 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { 1.5000E+01 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -1.299226E-10 -1.299225E-10 -1.366476E-10 -2.059599E-10 CZ -7.502087E-03 -7.502087E-03 -9.901986E-03 -1.222185E-02 CMX 3.987231E-10 3.987492E-10 6.668876E-10 6.946241E-10 CMY -4.898731E-03 -4.898731E-03 -6.364542E-03 -7.934161E-03 CMZ 8.584891E-11 8.596750E-11 1.202454E-10 1.436792E-10 ANGLEA CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -9.052926E-08 -9.052923E-08 -9.356474E-08 -1.382898E-07 CZ -5.040129E+00 -5.040129E+00 -6.537387E+00 -7.992975E+00 CMX 2.459049E-07 2.459198E-07 4.120568E-07 4.299212E-07 CMY -2.877635E+00 -2.877635E+00 -3.791211E+00 -4.776923E+00 CMZ 6.120506E-08 6.127953E-08 8.217181E-08 9.733981E-08 PITCH CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -2.032311E-07 -2.032310E-07 -1.898720E-07 -2.834348E-07 CZ -1.198419E+01 -1.198419E+01 -1.324751E+01 -1.683722E+01 CMX 4.787124E-07 4.787220E-07 6.819224E-07 7.736392E-07 CMY -9.632813E+00 -9.632813E+00 -1.035169E+01 -1.276509E+01 CMZ 1.120977E-07 1.123463E-07 1.328826E-07 1.711292E-07 URDD3 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 5.650741E-11 0.000000E+00 CZ 0.000000E+00 0.000000E+00 4.103939E-03 0.000000E+00 CMX 0.000000E+00 0.000000E+00 -2.388719E-10 0.000000E+00 CMY 0.000000E+00 0.000000E+00 2.820731E-03 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 -4.494630E-11 0.000000E+00 URDD5 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 1.122847E-09 0.000000E+00 CZ 0.000000E+00 0.000000E+00 8.055958E-02 0.000000E+00 CMX 0.000000E+00 0.000000E+00 -5.010415E-09 0.000000E+00 CMY 0.000000E+00 0.000000E+00 5.239135E-02 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 -9.285254E-10 0.000000E+00 ELEV CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -4.873205E-09 -4.873205E-09 -4.295436E-09 -5.451006E-09 CZ -1.037880E-01 -1.037880E-01 -1.599575E-01 -1.444930E-01 CMX 3.078125E-09 3.078583E-09 7.282612E-09 5.192063E-09 CMY 2.340457E-01 2.340457E-01 1.988105E-01 2.088675E-01 CMZ 6.640054E-09 6.640952E-09 6.962484E-09 7.085203E-09 SIDES CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -7.493702E-01 -7.493702E-01 -7.137761E-01 -7.576051E-01 CZ 1.927537E-07 1.927536E-07 2.877982E-07 3.301954E-07 CMX -1.354676E-01 -1.354676E-01 -1.477700E-01 -1.363622E-01 CMY 1.226018E-07 1.218731E-07 1.806978E-07 2.096502E-07 CMZ 1.900327E-01 1.900327E-01 1.774890E-01 1.929883E-01 YAW CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 7.217304E-01 7.217304E-01 6.687398E-01 7.260464E-01 CZ -7.310567E-08 -7.310565E-08 -9.128657E-08 -1.102763E-07 CMX 9.353673E-02 9.353673E-02 8.854351E-02 8.711862E-02 CMY -4.576293E-08 -4.524614E-08 -5.646639E-08 -6.888230E-08 CMZ -2.584813E-01 -2.584813E-01 -2.425677E-01 -2.609811E-01 ROLL CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY -1.516381E-01 -1.516381E-01 -1.633789E-01 -1.590035E-01 CZ 5.777496E-07 5.777496E-07 1.002999E-06 1.079415E-06 CMX -4.362884E-01 -4.362884E-01 -5.353552E-01 -4.651975E-01 CMY 3.642278E-07 3.617961E-07 6.268094E-07 6.827204E-07 CMZ -9.201576E-04 -9.201575E-04 -9.716505E-03 -2.064240E-03 URDD2 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 7.924410E-04 0.000000E+00 CZ 0.000000E+00 0.000000E+00 -2.779882E-10 0.000000E+00 CMX 0.000000E+00 0.000000E+00 1.503346E-04 0.000000E+00 CMY 0.000000E+00 0.000000E+00 -1.717070E-10 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 -2.422818E-04 0.000000E+00 URDD4 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 1.123470E-04 0.000000E+00 CZ 0.000000E+00 0.000000E+00 7.046271E-09 0.000000E+00 CMX 0.000000E+00 0.000000E+00 -3.891726E-03 0.000000E+00 CMY 0.000000E+00 0.000000E+00 4.531330E-09 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 -5.270858E-04 0.000000E+00 URDD6 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 -2.387614E-02 0.000000E+00 CZ 0.000000E+00 0.000000E+00 8.716039E-09 0.000000E+00 CMX 0.000000E+00 0.000000E+00 -4.560409E-03 0.000000E+00 CMY 0.000000E+00 0.000000E+00 5.405408E-09 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 6.942538E-03 0.000000E+00 AILERON CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 4.911802E-02 4.911802E-02 5.241855E-02 4.971820E-02 CZ -3.376502E-07 -3.376502E-07 -4.490562E-07 -4.774820E-07 CMX 2.803040E-01 2.803040E-01 3.064853E-01 2.662785E-01 CMY -2.574724E-07 -2.565203E-07 -3.250403E-07 -3.461600E-07 CMZ 5.166404E-03 5.166404E-03 7.268140E-03 2.985376E-03 RUDDER NX Nastran Aeroelastic CX Analysis User’s0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Guide 6-124 CY 2.297473E-01 2.297473E-01 1.926551E-01 2.195247E-01 CZ -1.595076E-09 -1.595073E-09 1.276250E-08 1.128636E-08 CMX 3.164924E-02 3.164924E-02 2.404623E-02 2.562781E-02

PAGE

Static Aeroelastic Analysis Sample Problems

INTERMEDIATE MATRIX ... HP0 COLUMN 1 -8.813170E-11 2.237854E-03 -5.660327E-10 -1.218842E-04 -2.451579E-11 INTERMEDIATE MATRIX ... HP COLUMN 1 -5.818967E-08 1.414439E+00 -3.489635E-07 -7.417409E-02 -1.609712E-08 COLUMN 2 -9.743544E-08 2.969788E+00 -5.736758E-07 -2.094978E-01 -2.610605E-08 COLUMN 3 3.281455E-11 5.562317E-03 2.026536E-10 -1.233582E-04 8.997147E-12 COLUMN 4 6.772485E-10 -1.540755E-03 4.252287E-09 -7.445705E-03 1.867415E-10 COLUMN 5 -3.466741E-09 -6.417216E-03 -5.491675E-09 2.293641E-03 -9.819316E-10 COLUMN 6 -1.529903E-01 -6.875273E-08 1.071376E-01 3.554970E-09 -2.518214E-02 COLUMN 7 1.536015E-01 2.205481E-08 -5.494779E-02 -1.111257E-09 3.605024E-02 COLUMN 8 1.071310E-02 -2.447038E-07 4.499267E-01 1.233983E-08 7.624566E-03 COLUMN 9 -5.622060E-03 6.714389E-11 -2.100419E-05 -3.355533E-12 -3.169009E-04 COLUMN 10 -1.395212E-04 -1.718798E-09 2.456579E-02 8.955151E-11 1.693952E-05 COLUMN 11 1.966925E-02 -2.092211E-09 2.716776E-03 1.057792E-10 1.675671E-02 COLUMN 12 -5.356136E-03 1.152103E-07 -2.441650E-01 -6.919825E-09 -3.753309E-03 COLUMN 13 7.398353E-02 -2.923241E-09 -1.039103E-02 1.238324E-10 1.980789E-02 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 1.907433E-03 PITCH -1.084202E-19 URDD3 -1.000000E+00 URDD5 0.000000E+00 ELEV 4.417495E-02 SIDES 1.959896E-09 YAW 3.874560E-09 ROLL 3.860787E-09 URDD2 0.000000E+00 URDD4 0.000000E+00 URDD6 1.615587E-27 AILERON 0.000000E+00 RUDDER 0.000000E+00 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 2.197457E-01 2.636949E+02 1000 LS 3.296252E+02 2.060158E+02 2 LS 2.134067E-02 2.560880E+01 1001 LS 3.201165E+01 2.000728E+01 3 LS 8.235436E-03 9.882523E+00 1002 LS 1.235340E+01 7.720877E+00 4 LS 6.941908E-03 8.330289E+00 1003 LS 1.041307E+01 6.508169E+00 5 LS 2.061283E-01 2.473539E+02 1004 LS 1.545956E+03 9.662225E+02 6 LS 1.982115E-02 2.378538E+01 1005 LS 1.486581E+02 9.291129E+01 7 LS 7.704427E-03 9.245313E+00 1006 LS 5.778298E+01 3.611436E+01 8 LS 6.705724E-03 8.046868E+00 1007 LS 5.029273E+01 3.143295E+01 9 LS -1.653892E-02 -1.984671E+01 1104 LS -1.240419E+02 -7.752621E+01 10 LS 7.864603E-03 9.437523E+00 1105 LS 5.898452E+01 3.686533E+01 11 LS 4.503795E-03 5.404554E+00 1106 LS 3.377847E+01 2.111155E+01 12 LS 1.776906E-03 2.132287E+00 1107 LS 1.332679E+01 8.329247E+00 13 LS 8.997116E-02 1.079654E+02 1108 LS 6.747838E+02 4.217399E+02 14 LS 1.303863E-02 1.564635E+01 1109 LS 9.778970E+01 6.111857E+01 15 LS 6.372170E-03 7.646604E+00 1110 LS 4.779128E+01 2.986956E+01 16 LS 3.041159E-03 3.649391E+00 1111 LS 2.280869E+01 1.425543E+01 17 LS 9.469719E-02 1.136366E+02 1112 LS 7.102289E+02 4.438931E+02 18 LS 2.907472E-02 3.488967E+01 1113 LS 2.180604E+02 1.362878E+02 19 LS 9.546882E-03 1.145626E+01 1114 LS 7.160162E+01 4.475101E+01 20 LS 4.849496E-03 5.819396E+00 1115 LS 3.637122E+01 2.273201E+01 21 LS 8.148860E-02 9.778632E+01 1116 LS 6.232680E+02 3.895425E+02 22 LS 2.826109E-02 3.391330E+01 1117 LS 2.161557E+02 1.350973E+02 23 LS 1.321945E-02 1.586334E+01 1118 LS 1.011094E+02 6.319334E+01 24 LS 5.626781E-03 6.752137E+00 1119 LS 4.303660E+01 2.689787E+01 25 LS 6.758305E-02 8.109966E+01 1120 LS 5.169109E+02 3.230693E+02 26 LS 2.871564E-02 3.445876E+01 1121 LS 2.196324E+02 1.372702E+02 27 LS 1.489998E-02 1.787998E+01 1122 LS 1.139630E+02 7.122684E+01 28 LS 6.808581E-03 8.170298E+00 1123 LS 5.207563E+01 3.254727E+01 29 LS 5.297777E-02 6.357333E+01 1124 LS 4.052021E+02 2.532513E+02 30 LS 2.394363E-02 2.873236E+01 1125 LS 1.831336E+02 1.144585E+02 31 LS 1.422081E-02 1.706497E+01 1126 LS 1.087683E+02 6.798019E+01 32 LS 7.345882E-03 8.815059E+00 1127 LS 5.618520E+01 3.511575E+01 33 LS 3.528650E-02 4.234381E+01 1128 LS 2.698899E+02 1.686812E+02 34 LS 1.595349E-02 1.914419E+01 1129 LS 1.220207E+02 7.626295E+01 35 LS 1.036819E-02 1.244183E+01 1130 LS 7.930141E+01 4.956339E+01 36 LS 6.222871E-03 7.467445E+00 1131 LS 4.759581E+01 2.974738E+01 37 LS 2.061283E-01 2.473539E+02 2000 LS 1.545956E+03 9.662225E+02 38 LS 1.982118E-02 2.378542E+01 2001 LS 1.486583E+02 9.291142E+01 39 LS 7.704393E-03 9.245272E+00 2002 LS 5.778272E+01 3.611420E+01 40 LS 6.705716E-03 8.046859E+00 2003 LS 5.029267E+01 3.143291E+01

NX Nastran Aeroelastic Analysis User’s Guide

6-125

Chapter 6

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131

6-126

LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* ZIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* YIB* ZSB* ZSB* ZSB* ZSB* ZSB* ZSB* ZSB* ZSB* ZSB* ZSB* ZSB*

Static Aeroelastic Analysis Sample Problems

2.197458E-01 2.134071E-02 8.235380E-03 6.941923E-03 3.528652E-02 1.595351E-02 1.036821E-02 6.222877E-03 5.297777E-02 2.394365E-02 1.422082E-02 7.345878E-03 6.758301E-02 2.871561E-02 1.489997E-02 6.808588E-03 8.148856E-02 2.826110E-02 1.321943E-02 5.626791E-03 9.469719E-02 2.907470E-02 9.546896E-03 4.849524E-03 8.997115E-02 1.303864E-02 6.372197E-03 3.041142E-03 -1.653893E-02 7.864612E-03 4.503803E-03 1.776905E-03 -1.501252E-09 3.040361E-09 2.393172E-09 6.758486E-10 -1.229602E-09 2.333110E-09 3.938139E-09 2.000437E-09 1.249291E-09 -4.946589E-09 6.511183E-09 4.136411E-09 1.349874E-02 -9.369577E-04 -1.349874E-02 9.369312E-04 1.236755E-02 2.627017E-02 1.044219E-01 -8.806521E-02 4.486084E-02 -1.320293E-01 -5.379935E-02 -2.621078E-02 -4.027994E-03 -2.357787E-03 1.977045E-03 3.425907E-03 -4.028021E-03 -2.357827E-03 1.976989E-03 3.425851E-03 -2.791346E-09 6.776578E-09 -1.548653E-08 5.712243E-07 1.987807E-08 3.456430E-07 1.699642E-07 -1.063246E-07 -1.948816E-02 -2.471328E-02 -2.343771E-02 -2.363333E-02 1.948814E-02 2.471327E-02 2.343768E-02 2.363331E-02 4.610739E-03 3.166118E-02 6.076573E-02 8.306995E-02 1.062085E-01 1.704039E-01 2.349322E-01 2.380018E-01 3.587926E-03 1.435171E-02 1.435171E-02

2.636949E+02 2.560886E+01 9.882456E+00 8.330308E+00 4.234382E+01 1.914421E+01 1.244185E+01 7.467453E+00 6.357333E+01 2.873237E+01 1.706498E+01 8.815054E+00 8.109961E+01 3.445874E+01 1.787996E+01 8.170306E+00 9.778627E+01 3.391332E+01 1.586332E+01 6.752150E+00 1.136366E+02 3.488964E+01 1.145628E+01 5.819428E+00 1.079654E+02 1.564636E+01 7.646636E+00 3.649371E+00 -1.984671E+01 9.437534E+00 5.404564E+00 2.132286E+00 -1.801503E-06 3.648433E-06 2.871806E-06 8.110183E-07 -1.475522E-06 2.799732E-06 4.725766E-06 2.400525E-06 1.499149E-06 -5.935907E-06 7.813420E-06 4.963693E-06 1.619849E+01 -1.124349E+00 -1.619849E+01 1.124317E+00 1.484106E+01 3.152420E+01 1.253063E+02 -1.056782E+02 5.383301E+01 -1.584351E+02 -6.455922E+01 -3.145294E+01 -4.833593E+00 -2.829345E+00 2.372454E+00 4.111089E+00 -4.833625E+00 -2.829393E+00 2.372386E+00 4.111021E+00 -3.349615E-06 8.131894E-06 -1.858383E-05 6.854691E-04 2.385368E-05 4.147716E-04 2.039571E-04 -1.275895E-04 -2.338579E+01 -2.965594E+01 -2.812525E+01 -2.836000E+01 2.338577E+01 2.965592E+01 2.812522E+01 2.835998E+01 5.532887E+00 3.799342E+01 7.291888E+01 9.968394E+01 1.274501E+02 2.044846E+02 2.819186E+02 2.856022E+02 4.305511E+00 1.722205E+01 1.722205E+01

NX Nastran Aeroelastic Analysis User’s Guide

2004 2005 2006 2007 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111 3500 3501 3600 3601 4000 4000 4001 4001 4002 4002 4003 4003 4004 4004 4005 4005 4006 4006 4007 4007 4510 4510 4511 4511 4512 4512 4513 4513 4610 4610 4611 4611 4612 4612 4613 4613

LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS LS YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB YSB ZSB

3.296252E+02 3.201171E+01 1.235332E+01 1.041309E+01 2.698900E+02 1.220209E+02 7.930154E+01 4.759586E+01 4.052021E+02 1.831337E+02 1.087683E+02 5.618517E+01 5.169107E+02 2.196322E+02 1.139629E+02 5.207570E+01 6.232676E+02 2.161558E+02 1.011092E+02 4.303668E+01 7.102290E+02 2.180603E+02 7.160173E+01 3.637143E+01 6.747837E+02 9.778978E+01 4.779148E+01 2.280857E+01 -1.240420E+02 5.898459E+01 3.377853E+01 1.332679E+01 -1.125939E-05 2.280271E-05 1.794879E-05 5.068864E-06 -9.222016E-06 1.749833E-05 2.953605E-05 1.500328E-05 9.369681E-06 -3.709942E-05 4.883387E-05 3.102308E-05 4.049623E+01 -2.810873E+00 -4.049623E+01 2.810794E+00 -4.918611E-05 1.874086E+01 -4.241869E-05 3.483097E+01 3.101547E-05 2.752336E+02 3.744381E-04 7.336660E+02 -3.092265E-04 2.818890E+02 1.870033E-04 2.297896E+02 -6.144900E-05 1.859019E+02 -7.878794E-05 4.293805E+01 5.453679E+00 1.407799E+00 1.715474E+01 5.035899E+00 2.048823E+00 3.964195E+00 1.437641E+00 -7.397157E+00 -5.453685E+00 1.407766E+00 -1.715474E+01 5.035889E+00 -2.048852E+00 3.964183E+00 -1.437648E+00 -7.397151E+00

2.060158E+02 2.000732E+01 7.720824E+00 6.508184E+00 1.686812E+02 7.626305E+01 4.956346E+01 2.974741E+01 2.532513E+02 1.144585E+02 6.798022E+01 3.511573E+01 3.230692E+02 1.372701E+02 7.122679E+01 3.254731E+01 3.895422E+02 1.350974E+02 6.319324E+01 2.689792E+01 4.438932E+02 1.362877E+02 4.475109E+01 2.273214E+01 4.217399E+02 6.111862E+01 2.986968E+01 1.425535E+01 -7.752623E+01 3.686537E+01 2.111158E+01 8.329241E+00 -7.037121E-06 1.425170E-05 1.121800E-05 3.168040E-06 -5.763761E-06 1.093646E-05 1.846003E-05 9.377050E-06 5.856051E-06 -2.318714E-05 3.052117E-05 1.938943E-05 2.531014E+01 -1.756796E+00 -2.531015E+01 1.756746E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00

Static Aeroelastic Analysis Sample Problems

132 ZSB* 3.587926E-03 4.305511E+00 133 ZSB* 3.587923E-03 4.305508E+00 134 ZSB* 1.435169E-02 1.722203E+01 135 ZSB* 1.435169E-02 1.722203E+01 136 ZSB* 3.587923E-03 4.305508E+00 137 YSB* -2.082073E-08 -2.498488E-05 138 YSB* -1.153673E-07 -1.384408E-04 139 YSB* -1.535662E-07 -1.842794E-04 140 YSB* -1.229189E-07 -1.475027E-04 141 YSB* -7.420095E-08 -8.904114E-05 142 YSB* -3.289821E-08 -3.947785E-05 143 YSB* 3.499495E-08 4.199394E-05 144 YSB* 1.110134E-07 1.332161E-04 145 YSB* -1.428826E-05 -1.714591E-02 146 YSB* -5.715151E-05 -6.858181E-02 147 YSB* -5.714999E-05 -6.857999E-02 148 YSB* -1.428712E-05 -1.714454E-02 149 YSB* 1.428725E-05 1.714470E-02 150 YSB* 5.715052E-05 6.858063E-02 151 YSB* 5.715204E-05 6.858245E-02 152 YSB* 1.428839E-05 1.714607E-02 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. * NOTE THAT, FOR BODY ELEMENTS, PRESSURES ARE NOT AVAILABLE. THE VALUES PRINTED REPRESENT DOUBLET STRENGTHS. PLEASE SEE THE HANDBOOK FOR AEROELASTICITY. D I S P L A C E M E N T V E C T O R POINT ID. TYPE T1 T2 T3 R1 R2 R3 88 G 0.0 0.0 2.066694E-04 -5.459292E-05 1.158922E-04 0.0 89 G 0.0 0.0 4.933557E-05 -4.350332E-05 6.648706E-05 0.0 90 G 0.0 0.0 0.0 0.0 0.0 0.0 91 G 0.0 0.0 4.933557E-05 4.350332E-05 6.648705E-05 0.0 92 G 0.0 0.0 2.066694E-04 5.459292E-05 1.158922E-04 0.0 97 G 0.0 -1.627291E-10 -7.360668E-03 0.0 -7.187539E-04 1.863641E-11 98 G 0.0 -1.447897E-11 -1.161003E-03 0.0 -4.223916E-04 7.202234E-12 99 G 4.896077E-20 -3.723963E-13 -9.832990E-04 1.888854E-12 3.988778E-04 1.261683E-12 100 G 1.600101E-19 1.287807E-10 -1.012783E-02 5.666561E-12 1.530052E-03 3.072896E-11 110 G -2.837356E-05 -1.786642E-05 -2.610124E-03 7.465275E-04 2.258455E-03 1.107073E-05 111 G -2.837356E-05 -4.554326E-05 3.036012E-03 7.465275E-04 2.258455E-03 1.107073E-05 112 G -2.837356E-05 9.810412E-06 -8.256260E-03 7.465275E-04 2.258455E-03 1.107073E-05 115 G -1.040624E-04 -6.305047E-05 9.420018E-03 1.143538E-03 2.660419E-03 1.892614E-05 120 G 2.517411E-03 -1.376084E-03 2.355001E-02 1.248269E-03 2.905261E-03 4.922319E-05 121 G 2.517411E-03 -1.499142E-03 3.081316E-02 1.248269E-03 2.905261E-03 4.922319E-05 122 G 2.517411E-03 -1.253026E-03 1.628686E-02 1.248269E-03 2.905261E-03 4.922319E-05 150 G -4.094846E-03 1.561312E-03 2.272429E-02 1.143553E-03 2.660969E-03 1.815946E-05 151 G -4.094846E-03 1.652271E-03 9.420020E-03 1.143553E-03 2.660625E-03 1.825637E-05 210 G -2.837325E-05 1.786650E-05 -2.610124E-03 -7.465275E-04 2.258454E-03 -1.107067E-05 211 G -2.837325E-05 4.554318E-05 3.036012E-03 -7.465275E-04 2.258454E-03 -1.107067E-05 212 G -2.837325E-05 -9.810176E-06 -8.256260E-03 -7.465275E-04 2.258454E-03 -1.107067E-05 215 G -1.040618E-04 6.305036E-05 9.420017E-03 -1.143537E-03 2.660419E-03 -1.892608E-05 220 G 2.517412E-03 1.376084E-03 2.355001E-02 -1.248269E-03 2.905261E-03 -4.922312E-05 221 G 2.517412E-03 1.499142E-03 3.081316E-02 -1.248269E-03 2.905261E-03 -4.922312E-05 222 G 2.517412E-03 1.253026E-03 1.628686E-02 -1.248269E-03 2.905261E-03 -4.922312E-05 250 G -4.094845E-03 -1.561312E-03 2.272429E-02 -1.143553E-03 2.660969E-03 -1.815940E-05 251 G -4.094845E-03 -1.652271E-03 9.420019E-03 -1.143553E-03 2.660625E-03 -1.825630E-05 310 G 7.658370E-03 1.936182E-10 -1.454966E-02 7.282316E-12 1.532231E-03 3.598105E-11 311 G 7.658370E-03 1.036656E-10 -1.071908E-02 7.282316E-12 1.532231E-03 3.598105E-11 312 G 7.658370E-03 2.835708E-10 -1.838024E-02 7.282316E-12 1.532231E-03 3.598105E-11 890 G 0.0 0.0 4.933557E-05 -4.350332E-05 6.648706E-05 0.0 910 G 0.0 0.0 4.933557E-05 4.350332E-05 6.648705E-05 0.0 F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 89 -7.741117E+02 0.0 2.622377E+03 0.0 -1.132163E+03 0.0 0.0 -4.117094E+03 90 3.250819E+03 0.0 7.625005E+03 0.0 -2.187093E+03 0.0 0.0 -8.310884E+03 91 7.625004E+03 0.0 3.250818E+03 0.0 2.187093E+03 0.0 0.0 8.310882E+03 92 2.622377E+03 0.0 -7.741116E+02 0.0 1.132163E+03 0.0 0.0 4.117093E+03 100-3.822030E+04 2.566290E-04-4.155522E+04-4.746479E-04 6.669843E+02 1.462554E-04 5.640281E-11 2.039962E-04 101-5.456968E-12-2.241588E-17-2.963621E+04 9.879128E-04 2.963621E+03-9.879128E-05 0.0 0.0 102-2.963621E+04 9.879128E-04-5.484206E+04 2.566332E-04 5.041170E+03 1.462559E-04 0.0 0.0 103-4.155522E+04-4.746479E-04-7.156216E+04-2.071325E-03 3.000694E+03 1.596677E-04 6.396444E-11 2.039962E-04 110 5.200714E+04-6.324351E+03 3.552312E+04-4.720463E+03 2.855118E+03-2.778015E+02-4.811666E+02 1.115231E+04 115 3.282450E+04-4.720463E+03 1.435679E+04-3.116575E+03 3.198701E+03-2.778016E+02-4.811666E+02 6.478144E+03 120 1.428939E+04-2.075387E+03 3.896671E+03-4.824830E+02 1.773664E+03-2.718516E+02-1.706641E+02 6.933296E+03 150 1.455192E-11 2.910383E-11 3.435822E+01-1.116298E+02-6.871643E+00 2.232595E+01 2.910383E-11 -2.728484E-12 151-3.435822E+01-5.820766E-11-3.435822E+01 5.798500E+01-8.731149E-11-3.865667E+01-3.010736E+00 1.116298E+02 210 5.200714E+04 6.324349E+03 3.552311E+04 4.720462E+03 2.855118E+03 2.778014E+02-4.811665E+02 -1.115231E+04 215 3.282450E+04 4.720462E+03 1.435679E+04 3.116574E+03 3.198700E+03 2.778015E+02-4.811664E+02 -6.478143E+03 220 1.428939E+04 2.075387E+03 3.896670E+03 4.824830E+02 1.773664E+03 2.718516E+02-1.706641E+02 -6.933294E+03 250-2.910383E-11 2.910383E-11 3.435807E+01 1.116297E+02-6.871614E+00-2.232594E+01 0.0 5.456968E-12 251-3.435807E+01 1.746230E-10-3.435807E+01-5.798510E+01-5.820766E-11 3.865673E+01-3.010688E+00 -1.116297E+02 310-2.386749E+02-1.089980E-03 4.999999E+01-1.339079E-04-4.999997E+01-1.655967E-04-8.660255E+01 2.319354E-04 S T R E S S E S I N B A R E L E M E N T S ( C B A R ) ELEMENT SA1 SA2 SA3 SA4 AXIAL SA-MAX SA-MIN M.S.-T ID. SB1 SB2 SB3 SB4 STRESS SB-MAX SB-MIN M.S.-C 0 89 2.229443E+03 2.229443E+03 -2.229443E+03 -2.229443E+03 0.0 2.229443E+03 -2.229443E+03 -7.552450E+03 -7.552450E+03 7.552450E+03 7.552450E+03 7.552450E+03 -7.552450E+03 0 90 -9.362364E+03 -9.362364E+03 9.362364E+03 9.362364E+03 0.0 9.362364E+03 -9.362364E+03 -2.196003E+04 -2.196003E+04 2.196003E+04 2.196003E+04 2.196003E+04 -2.196003E+04 0 91 -2.196003E+04 -2.196003E+04 2.196003E+04 2.196003E+04 0.0 2.196003E+04 -2.196003E+04 -9.362363E+03 -9.362363E+03 9.362363E+03 9.362363E+03 9.362363E+03 -9.362363E+03 0 92 -7.552450E+03 -7.552450E+03 7.552450E+03 7.552450E+03 0.0 7.552450E+03 -7.552450E+03

NX Nastran Aeroelastic Analysis User’s Guide

6-127

Chapter 6

0 100 0 101 0 102 0 103 0 110 0 115 0 120 0 150 0 151 0 210 0 215 0 220 0 250 0 251 0 310

6-128

2.229443E+03 1.100745E+05 1.196791E+05 1.571615E-11 8.535234E+04 8.535234E+04 1.579452E+05 1.196791E+05 2.060992E+05 -1.402941E+05 -9.522595E+04 -8.745391E+04 -3.667271E+04 -3.804039E+04 -1.049869E+04 -8.556529E-11 6.849292E+01 9.895172E+01 1.197422E+01 -1.592672E+05 -1.093873E+05 -1.016153E+05 -4.602245E+04 -4.426655E+04 -1.194614E+04 4.016334E-11 -2.663958E+02 9.895129E+01 1.859290E+02 6.873859E+02 -1.439999E+02

Static Aeroelastic Analysis Sample Problems

2.229443E+03 1.100745E+05 1.196791E+05 1.571600E-11 8.535234E+04 8.535234E+04 1.579452E+05 1.196791E+05 2.060991E+05 -1.592672E+05 -1.093873E+05 -1.016153E+05 -4.602244E+04 -4.426656E+04 -1.194614E+04 1.746203E-12 -2.663964E+02 9.895172E+01 1.859292E+02 -1.402941E+05 -9.522594E+04 -8.745391E+04 -3.667272E+04 -3.804039E+04 -1.049869E+04 1.274748E-10 6.849322E+01 9.895129E+01 1.197364E+01 6.873826E+02 -1.440003E+02

-2.229443E+03 -1.100745E+05 -1.196791E+05 -1.571600E-11 -8.535234E+04 -8.535234E+04 -1.579452E+05 -1.196791E+05 -2.060991E+05 1.592672E+05 1.093873E+05 1.016153E+05 4.602244E+04 4.426656E+04 1.194614E+04 -1.746203E-12 2.663964E+02 -9.895172E+01 -1.859292E+02 1.402941E+05 9.522594E+04 8.745391E+04 3.667272E+04 3.804039E+04 1.049869E+04 -1.274748E-10 -6.849322E+01 -9.895129E+01 -1.197364E+01 -6.873826E+02 1.440003E+02

NX Nastran Aeroelastic Analysis User’s Guide

-2.229443E+03 -1.100745E+05 -1.196791E+05 -1.571615E-11 -8.535234E+04 -8.535234E+04 -1.579452E+05 -1.196791E+05 -2.060992E+05 1.402941E+05 9.522595E+04 8.745391E+04 3.667271E+04 3.804039E+04 1.049869E+04 8.556529E-11 -6.849292E+01 -9.895172E+01 -1.197422E+01 1.592672E+05 1.093873E+05 1.016153E+05 4.602245E+04 4.426655E+04 1.194614E+04 -4.016334E-11 2.663958E+02 -9.895129E+01 -1.859290E+02 -6.873859E+02 1.439999E+02

1.410070E-11 0.0 0.0 1.599111E-11 -3.207777E+02 -3.207777E+02 -1.137760E+02 1.940255E-11 -2.007157E+00 -3.207776E+02 -3.207776E+02 -1.137760E+02 0.0 -2.007125E+00 -5.773503E+01

2.229443E+03 1.100745E+05 1.196791E+05 1.571615E-11 8.535234E+04 8.535234E+04 1.579452E+05 1.196791E+05 2.060992E+05 1.589464E+05 1.090666E+05 1.012945E+05 4.570166E+04 4.415278E+04 1.183237E+04 1.049678E-10 2.663964E+02 9.694456E+01 1.839221E+02 1.589464E+05 1.090665E+05 1.012945E+05 4.570167E+04 4.415278E+04 1.183236E+04 1.274748E-10 2.663958E+02 9.694417E+01 1.839218E+02 6.296508E+02 8.626524E+01

-2.229443E+03 -1.100745E+05 -1.196791E+05 -1.571615E-11 -8.535234E+04 -8.535234E+04 -1.579452E+05 -1.196791E+05 -2.060992E+05 -1.595880E+05 -1.097081E+05 -1.019361E+05 -4.634322E+04 -4.438034E+04 -1.205992E+04 -6.616274E-11 -2.663964E+02 -1.009589E+02 -1.879364E+02 -1.595880E+05 -1.097081E+05 -1.019361E+05 -4.634322E+04 -4.438033E+04 -1.205992E+04 -1.274748E-10 -2.663958E+02 -1.009584E+02 -1.879361E+02 -7.451209E+02 -2.017353E+02

Static Aeroelastic Analysis Sample Problems

Listing 6-25. Output for FSW Airplane with Bodies

6.8 Unit Solutions for Loadings of the FSW Airplane (Examples HA144GA and HA144GB) The preceding examples of static aeroelastic analysis all result in trimmed solutions. For some design studies, it may be desirable to separate the trimmed loading into the unit solutions generated by each trim variable. These solutions are available from NX Nastran but require two different calculations, one for the initial incidence distribution and the other for the trim variables. The first calculation provides the solution for the static incidence distribution {wgj } and any experimental pressure input {fej } (the DMI entries W2GJ and FA2J, respectively) and requires a restrained vehicle, i.e., the SUPORT entry must be replaced by an SPC entry. As a consequence of this restraint, stability derivatives are not calculated, but all of the user-requested outputs will be found: e.g., aerodynamic pressures and forces, structural deflections, forces, etc. The stability derivatives can be obtained in the second calculation. This first calculation may also include any of the aerodynamic trim variables such as angle of attack, but it cannot include any of the acceleration trim variables such as a load factor. The second calculation requires a DMAP ALTER, the SUPORT entry, and one TRIM entry. If there are additional requested outputs (e.g., pressures), the Case Control Section must also include as many TRIM Subcases as there are trim variables. The only direct matrix input included in the unit trim variable analysis is the empirical correction factors [Wkk ] (the DMI entry WKK). Stability derivatives are obtained only for the first trim condition (if there is more than one) before the DMAP alter controls the solution. As many solutions are obtained as there are trim variables, and the solutions are for a unit value of each trim variable in sequence. Example HA144A is modified to demonstrate the two calculations. In the first modified file, Example HA144GA illustrates the replacement of SUPORT,90,35 by SPC1,90,35 and obtains three subcase solutions. Subcase 1 analyzes the case of the initial incidence of 0.1 deg on the wing with all trim variables set to zero on the TRIM 1 entry. Subcase 2 assumes no initial incidence and all trim variables set to zero except for the angle of attack, ANGLEA = 1.0. Subcase 3 also assumes no incidence and all trim variables are set to zero except for the canard rotation, ELEV = 1.0. The output from Subcases 2 and 3 should be the same as the corresponding solutions from our second ALTERed calculation, as will be seen. The Executive and Case Control Sections of Example HA144GA are shown in Listing 6-26 and the Sorted Bulk Data entries are in Listing 6-27. The output of aerodynamic pressures and forces, and structural deflections, forces, and stresses follows in Listing 6-28. In the second modified file, Example HA144GB illustrates the unit solutions for the five trim variables: angle of attack, pitch rate, normal acceleration, pitching acceleration, and canard rotation. Only minor modifications to Example HA144GA are required to obtain the input file for HA144GB. The Executive Control Section, containing the DMAP ALTER, and the Case Control Section are shown in Listing 6-29, the Sorted Bulk Data are shown in Listing 6-30. The Case Control Section contains five dummy TRIM subcases, giving a total of five subcases, in order also to obtain the unit solutions for structural deflections, element forces and stresses. The output is contained in Listing 6-8 and begins with the first subcase stability derivatives that are in agreement with Example HA144A (see Listing 6-3, Subcase 2). Next is the matrix FFAJ, the aerodynamic box pressures. The first column is for the unit angle of attack and the last column is for the unit canard rotation; these agree with Subcases 2 and 3, respectively, of

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Example HA144GA (Listing 6-28), as anticipated above. The next matrix, PAK, contains the lift and moment on each aerodynamic box, and again the first and last column are in agreement with HA144GA. This is followed by more standard output for Subcase 1 and then the additional requested output of the structural deflections and element forces and stresses for the five unit trim variables. These unit solutions can be output from NX Nastran in the OUTPUT4 format for use in subsequent postprocessors.

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N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA144GA TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO CEND EXAMPLE HA144GA: 30 DEG FWD SWEPT WING WITH CANARD PAGE 2 ISOLATED CAMBER SOLUTION PLUS ANGLEA AND CANARD ROTATIONS HALF-SPAN MODEL, STATIC SYMMETRIC LOADING C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144GA: 30 DEG FWD SWEPT WING WITH CANARD 2 SUBTI = ISOLATED CAMBER SOLUTION PLUS ANGLEA AND CANARD ROTATIONS 3 LABEL = HALF-SPAN MODEL, STATIC SYMMETRIC LOADING 4 ECHO = BOTH 5 SPC = 1 $ SYMMETRIC CONSTRAINTS 6 DISP = ALL $ PRINT ALL DISPLACEMENTS 7 STRESS = ALL $ PRINT ALL STRESSES 8 FORCE = ALL $ PRINT ALL FORCES 9 AEROF = ALL $ PRINT ALL AERODYNAMIC FORCES 10 APRES = ALL $ PRINT ALL AERODYNAMIC PRESSURES 11 SUBCASE 1 12 TRIM = 1 $ INTERCEPT SOLUTION 13 SUBCASE 2 14 TRIM = 2 $ ANGLEA SOLUTION 15 SUBCASE 3 16 TRIM = 3 $ ELEV SOLUTION 17 BEGIN BULK

Listing 6-26. Input File for Camber/Twist Loading of Example HA144GA EXAMPLE HA144GA: 30 DEG FWD SWEPT WING WITH CANARD PAGE ISOLATED CAMBER SOLUTION PLUS ANGLEA AND CANARD ROTATIONS HALF-SPAN MODEL, STATIC SYMMETRIC LOADING CARD COUNT 12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758-

11

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . AELIST 1000 1000 THRU 1007 AEROS 1 100 10.0 40.0 200.0 1 AESTAT 501 ANGLEA AESTAT 502 PITCH AESTAT 503 URDD3 AESTAT 504 URDD5 AESURF 505 ELEV 1 1000 CAERO1 1000 1000 2 4 1 +CAC +CAC 10. 0. 0. 10. 10. 5. 0. 10. CAERO1 1100 1000 8 4 1 +CAW +CAW 25. 0. 0. 10. 13.45299+20. 0. 10. CBAR 100 100 90 99 0. 0. 1. CBAR 101 100 97 98 0. 0. 1. CBAR 102 100 98 90 0. 0. 1. CBAR 103 100 99 100 0. 0. 1. CBAR 110 101 100 110 0. 0. 1. CBAR 120 101 110 120 0. 0. 1. CONM2 97 97 0 1500.0 CONM2 98 98 0 1500.0 CONM2 99 99 0 1500.0 CONM2 100 100 0 1500.0 CONM2 111 111 0 600.0 CONM2 112 112 0 400.0 CONM2 121 121 0 600.0 CONM2 122 122 0 400.0 CORD2R 1 0 12.5 0. 0. 12.5 0. 10. +CRD1 +CRD1 20. 0. 0. CORD2R 2 0 30. 0. 0. 30. 0. 10. +CRD2 +CRD2 38.66025+5.0 0. CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 -10.0 +CRD100 +CRD100 0.0 0.0 0.0 DMI FA2J 0 2 1 0 40 3 DMI FA2J 1 1 0.0 THRU 40 DMI FA2J 2 1 0.0 THRU 40 DMI FA2J 3 1 0.0 THRU 40 DMI W2GJ 0 2 1 0 40 3 DMI W2GJ 1 9 .0017453THRU 40 DMI W2GJ 2 9 .0 THRU 40 DMI W2GJ 3 9 .0 THRU 40 DMI WKK 0 3 1 0 80 1 DMI WKK 1 1 1.0 THRU 80 GRID 90 15. 0. 0. GRID 97 0. 0. 0. GRID 98 10. 0. 0. GRID 99 20. 0. 0. GRID 100 30. 0. 0. GRID 110 27.11325+5. 0. GRID 111 24.61325+5. 0. GRID 112 29.61325+5. 0. GRID 120 21.33975+15. 0. GRID 121 18.83975+15. 0. GRID 122 23.83975+15. 0. MAT1 1 1.44+9 5.40+8 OMIT1 4 110 120 PAERO1 1000 PARAM AUNITS .031081 6-131 NX Nastran Aeroelastic Analysis User’s Guide PARAM GRDPNT 90 PARAM WTMASS .031081

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59PBAR 100 1 2.0 .173611 0.15 0.5 60+PB1 1.0 1.0 1.0 -1.0 -1.0 1.0 -1.0 61+PB2 0.0 62PBAR 101 1 1.5 0.173611+2.0 0.462963 EXAMPLE HA144GA: 30 DEG FWD SWEPT WING WITH CANARD ISOLATED CAMBER SOLUTION PLUS ANGLEA AND CANARD ROTATIONS HALF-SPAN MODEL, STATIC SYMMETRIC LOADING S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 63+PB3 0.5 3.0 0.5 -3.0 -0.5 3.0 -0.5 64+PB4 0.0 65RBAR 111 110 111 123456 66RBAR 112 110 112 123456 67RBAR 121 120 121 123456 68RBAR 122 120 122 123456 69SET1 1000 98 99 70SET1 1100 99 100 111 112 121 122 71SPC1 1 35 90 72SPC1 1 246 97 98 99 100 73SPC1 1 1246 90 74SPLINE2 1501 1000 1000 1007 1000 0. 1. 75+SPC 1. -1. 76SPLINE2 1601 1100 1100 1131 1100 0. 1. 77+SPW -1. -1. 78TRIM 1 0.9 1200.0 ANGLEA 0.0 PITCH 0.0 79+TR1 URDD3 0.0 URDD5 0.0 ELEV 0.0 80TRIM 2 0.9 1200.0 ANGLEA 1.0 PITCH 0.0 81+TR2 URDD3 0.0 URDD5 0.0 ELEV 0.0 82TRIM 3 0.9 1200.0 ANGLEA 0.0 PITCH 0.0 83+TR3 URDD3 0.0 URDD5 0.0 ELEV 1.0 ENDDATA TOTAL COUNT= 84

-1.0

+PB1 +PB2

PAGE

+PB3 12

.. 9 -3.0

.. 10 +PB4

1

+SPC

2

+SPW

.

+TR1 +TR2 +TR3

Listing 6-27. Sorted Bulk Data for Camber/Twist Loading of Example HA144GA EXAMPLE HA144GA: 30 DEG FWD SWEPT WING WITH CANARD PAGE 17 ISOLATED CAMBER SOLUTION PLUS ANGLEA AND CANARD ROTATIONS HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 1 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 0.000000E+00 PITCH 0.000000E+00 URDD3 0.000000E+00 URDD5 0.000000E+00 ELEV 0.000000E+00 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 8.645258E-04 1.037431E+00 1000 LS 6.483944E+00 4.052465E+00 2 LS 9.342593E-04 1.121111E+00 1001 LS 7.006945E+00 4.379340E+00 3 LS 1.551324E-03 1.861588E+00 1002 LS 1.163493E+01 7.271830E+00 4 LS 2.427776E-03 2.913331E+00 1003 LS 1.820832E+01 1.138020E+01 5 LS 6.718651E-04 8.062381E-01 1004 LS 5.038988E+00 3.149368E+00 6 LS 7.300189E-04 8.760228E-01 1005 LS 5.475142E+00 3.421964E+00 7 LS 1.255685E-03 1.506822E+00 1006 LS 9.417638E+00 5.886024E+00 8 LS 2.173061E-03 2.607673E+00 1007 LS 1.629796E+01 1.018622E+01 9 LS 3.140397E-02 3.768477E+01 1100 LS 2.355298E+02 1.472061E+02 10 LS 8.190622E-03 9.828746E+00 1101 LS 6.142966E+01 3.839354E+01 11 LS 3.100009E-03 3.720011E+00 1102 LS 2.325007E+01 1.453130E+01 12 LS 1.090936E-03 1.309123E+00 1103 LS 8.182018E+00 5.113761E+00 13 LS 2.808250E-02 3.369899E+01 1104 LS 2.106187E+02 1.316367E+02 14 LS 1.036211E-02 1.243453E+01 1105 LS 7.771582E+01 4.857239E+01 15 LS 4.421213E-03 5.305456E+00 1106 LS 3.315910E+01 2.072444E+01 16 LS 1.355177E-03 1.626212E+00 1107 LS 1.016382E+01 6.352389E+00 17 LS 2.980389E-02 3.576467E+01 1108 LS 2.235292E+02 1.397057E+02 18 LS 1.167771E-02 1.401325E+01 1109 LS 8.758281E+01 5.473925E+01 19 LS 5.542692E-03 6.651231E+00 1110 LS 4.157019E+01 2.598137E+01 20 LS 1.428168E-03 1.713801E+00 1111 LS 1.071126E+01 6.694537E+00 21 LS 2.644349E-02 3.173219E+01 1112 LS 1.983262E+02 1.239538E+02 22 LS 1.156806E-02 1.388167E+01 1113 LS 8.676043E+01 5.422527E+01 23 LS 6.411158E-03 7.693390E+00 1114 LS 4.808369E+01 3.005231E+01 24 LS 2.861865E-03 3.434238E+00 1115 LS 2.146399E+01 1.341499E+01 25 LS 2.339144E-02 2.806973E+01 1116 LS 1.754358E+02 1.096474E+02 26 LS 1.060380E-02 1.272457E+01 1117 LS 7.952853E+01 4.970533E+01 27 LS 6.417695E-03 7.701234E+00 1118 LS 4.813271E+01 3.008295E+01 28 LS 3.311539E-03 3.973847E+00 1119 LS 2.483654E+01 1.552284E+01 29 LS 2.021148E-02 2.425377E+01 1120 LS 1.515861E+02 9.474129E+01 30 LS 9.227427E-03 1.107291E+01 1121 LS 6.920570E+01 4.325356E+01 31 LS 5.903500E-03 7.084200E+00 1122 LS 4.427625E+01 2.767266E+01 32 LS 3.336834E-03 4.004201E+00 1123 LS 2.502626E+01 1.564141E+01 33 LS 1.644886E-02 1.973863E+01 1124 LS 1.233665E+02 7.710404E+01 34 LS 7.445403E-03 8.934484E+00 1125 LS 5.584053E+01 3.490033E+01 35 LS 4.959068E-03 5.950882E+00 1126 LS 3.719301E+01 2.324563E+01 36 LS 3.048317E-03 3.657980E+00 1127 LS 2.286238E+01 1.428899E+01 37 LS 1.131609E-02 1.357930E+01 1128 LS 8.487065E+01 5.304416E+01 38 LS 4.946161E-03 5.935393E+00 1129 LS 3.709621E+01 2.318513E+01 39 LS 3.392896E-03 4.071475E+00 1130 LS 2.544672E+01 1.590420E+01 40 LS 2.313882E-03 2.776659E+00 1131 LS 1.735412E+01 1.084632E+01 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, NX Nastran Aeroelastic Analysis User’s Guide 6-132

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ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT,

YSB = Y SLENDER BODY ELEMENT.

HALF-SPAN MODEL, STATIC SYMMETRIC LOADING A E R O S T A T I C

SUBCASE 2 D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 1.000000E+00 PITCH 0.000000E+00 URDD3 0.000000E+00 URDD5 0.000000E+00 ELEV 0.000000E+00 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 7.638809E+00 9.166570E+03 1000 LS 5.729107E+04 3.580691E+04 2 LS 1.767498E+00 2.120998E+03 1001 LS 1.325624E+04 8.285148E+03 3 LS 1.290209E+00 1.548251E+03 1002 LS 9.676568E+03 6.047855E+03 4 LS 1.606494E+00 1.927792E+03 1003 LS 1.204870E+04 7.530439E+03 5 LS 6.041941E+00 7.250330E+03 1004 LS 4.531456E+04 2.832160E+04 6 LS 1.298661E+00 1.558393E+03 1005 LS 9.739958E+03 6.087474E+03 7 LS 1.019578E+00 1.223494E+03 1006 LS 7.646835E+03 4.779272E+03 8 LS 1.433167E+00 1.719800E+03 1007 LS 1.074875E+04 6.717970E+03 9 LS 1.291075E+01 1.549290E+04 1100 LS 9.683061E+04 6.051913E+04 10 LS 3.628104E+00 4.353725E+03 1101 LS 2.721078E+04 1.700674E+04 11 LS 1.392011E+00 1.670413E+03 1102 LS 1.044008E+04 6.525054E+03 12 LS 4.928862E-01 5.914635E+02 1103 LS 3.696647E+03 2.310404E+03 13 LS 1.345262E+01 1.614314E+04 1104 LS 1.008946E+05 6.305914E+04 14 LS 4.678058E+00 5.613669E+03 1105 LS 3.508543E+04 2.192839E+04 15 LS 2.022729E+00 2.427275E+03 1106 LS 1.517047E+04 9.481546E+03 16 LS 5.607044E-01 6.728453E+02 1107 LS 4.205283E+03 2.628302E+03 17 LS 2.084907E+01 2.501889E+04 1108 LS 1.563680E+05 9.773002E+04 18 LS 6.215158E+00 7.458190E+03 1109 LS 4.661369E+04 2.913355E+04 19 LS 2.683741E+00 3.220489E+03 1110 LS 2.012806E+04 1.258004E+04 20 LS 5.237615E-01 6.285137E+02 1111 LS 3.928211E+03 2.455132E+03 21 LS 1.768373E+01 2.122048E+04 1112 LS 1.326280E+05 8.289248E+04 22 LS 7.104998E+00 8.525997E+03 1113 LS 5.328748E+04 3.330468E+04 23 LS 3.455575E+00 4.146690E+03 1114 LS 2.591681E+04 1.619801E+04 24 LS 1.429336E+00 1.715203E+03 1115 LS 1.072002E+04 6.700013E+03 25 LS 1.517674E+01 1.821209E+04 1116 LS 1.138256E+05 7.114098E+04 26 LS 6.764285E+00 8.117142E+03 1117 LS 5.073214E+04 3.170758E+04 27 LS 3.771950E+00 4.526340E+03 1118 LS 2.828963E+04 1.768102E+04 28 LS 1.792641E+00 2.151169E+03 1119 LS 1.344481E+04 8.403006E+03 29 LS 1.286854E+01 1.544225E+04 1120 LS 9.651405E+04 6.032128E+04 30 LS 5.904088E+00 7.084905E+03 1121 LS 4.428066E+04 2.767541E+04 31 LS 3.638149E+00 4.365779E+03 1122 LS 2.728612E+04 1.705383E+04 32 LS 1.926708E+00 2.312049E+03 1123 LS 1.445031E+04 9.031441E+03 33 LS 1.034013E+01 1.240815E+04 1124 LS 7.755095E+04 4.846934E+04 34 LS 4.739940E+00 5.687928E+03 1125 LS 3.554955E+04 2.221847E+04 35 LS 3.116378E+00 3.739654E+03 1126 LS 2.337284E+04 1.460802E+04 36 LS 1.840009E+00 2.208011E+03 1127 LS 1.380007E+04 8.625043E+03 37 LS 7.047599E+00 8.457119E+03 1128 LS 5.285699E+04 3.303562E+04 38 LS 3.130690E+00 3.756828E+03 1129 LS 2.348017E+04 1.467511E+04 39 LS 2.145972E+00 2.575166E+03 1130 LS 1.609479E+04 1.005924E+04 40 LS 1.430286E+00 1.716344E+03 1131 LS 1.072715E+04 6.704467E+03 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 3 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 0.000000E+00 PITCH 0.000000E+00 URDD3 0.000000E+00 URDD5 0.000000E+00 ELEV 1.000000E+00 A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC PRES. AERODYNAMIC GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1 LS 7.143464E+00 8.572156E+03 1000 LS 5.357598E+04 3.348498E+04 2 LS 1.232198E+00 1.478638E+03 1001 LS 9.241486E+03 5.775929E+03 3 LS 4.013512E-01 4.816215E+02 1002 LS 3.010134E+03 1.881334E+03 4 LS 2.154573E-01 2.585487E+02 1003 LS 1.615929E+03 1.009956E+03 5 LS 5.656984E+00 6.788381E+03 1004 LS 4.242738E+04 2.651712E+04 6 LS 8.803840E-01 1.056461E+03 1005 LS 6.602880E+03 4.126800E+03 7 LS 3.001114E-01 3.601337E+02 1006 LS 2.250836E+03 1.406772E+03 8 LS 1.880739E-01 2.256887E+02 1007 LS 1.410554E+03 8.815964E+02 9 LS -5.082705E+00 -6.099246E+03 1100 LS -3.812029E+04 -2.382518E+04 10 LS -1.064855E+00 -1.277826E+03 1101 LS -7.986414E+03 -4.991508E+03 11 LS -3.841929E-01 -4.610315E+02 1102 LS -2.881447E+03 -1.800905E+03 12 LS -1.321844E-01 -1.586213E+02 1103 LS -9.913829E+02 -6.196143E+02 13 LS -2.637738E+00 -3.165285E+03 1104 LS -1.978303E+04 -1.236439E+04 14 LS -1.259093E+00 -1.510912E+03 1105 LS -9.443198E+03 -5.901999E+03 15 LS -5.104817E-01 -6.125781E+02 1106 LS -3.828613E+03 -2.392884E+03 16 LS -2.157675E-01 -2.589211E+02 1107 LS -1.618256E+03 -1.011410E+03 17 LS 3.772414E+00 4.526896E+03 1108 LS 2.829311E+04 1.768319E+04

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18 LS -4.757876E-01 -5.709451E+02 1109 LS -3.568407E+03 19 LS -4.920412E-01 -5.904494E+02 1110 LS -3.690309E+03 20 LS -2.945322E-01 -3.534386E+02 1111 LS -2.208991E+03 21 LS 2.532474E+00 3.038969E+03 1112 LS 1.899356E+04 22 LS 4.768783E-01 5.722540E+02 1113 LS 3.576588E+03 23 LS -2.178098E-01 -2.613718E+02 1114 LS -1.633574E+03 24 LS -2.104192E-01 -2.525031E+02 1115 LS -1.578144E+03 25 LS 1.774211E+00 2.129053E+03 1116 LS 1.330658E+04 26 LS 6.886503E-01 8.263804E+02 1117 LS 5.164877E+03 27 LS 9.482024E-02 1.137843E+02 1118 LS 7.111518E+02 28 LS -1.047628E-01 -1.257154E+02 1119 LS -7.857213E+02 29 LS 1.288023E+00 1.545628E+03 1120 LS 9.660174E+03 30 LS 6.170731E-01 7.404877E+02 1121 LS 4.628048E+03 31 LS 2.556365E-01 3.067638E+02 1122 LS 1.917273E+03 32 LS 1.481013E-02 1.777215E+01 1123 LS 1.110759E+02 33 LS 9.154637E-01 1.098557E+03 1124 LS 6.865978E+03 34 LS 4.739670E-01 5.687604E+02 1125 LS 3.554753E+03 35 LS 2.749938E-01 3.299926E+02 1126 LS 2.062454E+03 36 LS 9.342295E-02 1.121075E+02 1127 LS 7.006721E+02 37 LS 5.638503E-01 6.766204E+02 1128 LS 4.228878E+03 38 LS 2.967008E-01 3.560410E+02 1129 LS 2.225256E+03 39 LS 2.019532E-01 2.423438E+02 1130 LS 1.514649E+03 40 LS 1.045072E-01 1.254087E+02 1131 LS 7.838041E+02 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE D I S P L A C E M E N T POINT ID. TYPE T1 T2 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T3 0.0 2.337642E-05 5.844105E-06 6.539142E-04 2.312514E-03 4.003543E-03 4.968655E-03 3.038431E-03 1.244762E-02 1.406487E-02 1.083038E-02

D I S P L A C E M E N T POINT ID. TYPE T1 T2 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING D I S P L A POINT ID. TYPE T1 T2 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T3 0.0 8.257084E-02 2.064271E-02 3.520744E-01 1.070061E+00 2.231179E+00 2.935486E+00 1.526873E+00 7.741362E+00 8.863285E+00 6.619438E+00 C E M E N T T3 0.0 6.917690E-02 1.729423E-02 -2.259708E-02 -2.549343E-01 -6.271992E-02 8.860861E-02 -2.140484E-01 6.092786E-01 8.045742E-01 4.139830E-01

R1 R2 0.0 0.0 0.0 1.753231E-06 0.0 1.753231E-06 0.0 -2.208217E-04 0.0 4.265358E-05 3.714096E-04 3.860448E-04 3.714096E-04 3.860448E-04 3.714096E-04 3.860448E-04 6.134597E-04 6.468976E-04 6.134597E-04 6.468976E-04 6.134597E-04 6.468976E-04 SUBCASE 2

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

V E C T O R R1 0.0 0.0 0.0 0.0 0.0 2.300923E-01 2.300923E-01 2.300923E-01 3.824708E-01 3.824708E-01 3.824708E-01

R2 0.0 6.192812E-03 6.192812E-03 -1.170423E-01 6.436867E-02 2.817225E-01 2.817225E-01 2.817225E-01 4.487695E-01 4.487695E-01 4.487695E-01 SUBCASE 3

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

V E C T O R R1 0.0 0.0 0.0 0.0 0.0 1.728677E-02 1.728677E-02 1.728677E-02 3.097839E-02 3.097839E-02 3.097839E-02

R2 R3 0.0 0.0 5.188268E-03 0.0 5.188268E-03 0.0 9.481348E-03 0.0 3.992956E-02 0.0 6.053141E-02 0.0 6.053141E-02 0.0 6.053141E-02 0.0 7.811824E-02 0.0 7.811824E-02 0.0 7.811824E-02 0.0 SUBCASE 1

( C B A R ) AXIAL FORCE TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.835926E+03 0.0 2.270722E+03 SUBCASE 2

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE 100 9.420228E+06 0.0 2.283996E+06 0.0 1.427247E+06 0.0 0.0 101 -1.455192E-11 0.0 2.182787E-11 0.0 -3.637979E-12 0.0 0.0

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V E C T O R

F O R C E S I N B A R E L E M E N T S ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 100 1.715267E+04 0.0 4.929497E+03 0.0 2.444634E+03 0.0 101 0.0 0.0 0.0 0.0 0.0 0.0 102 -7.105427E-15 0.0 1.753230E+02 0.0 -3.506461E+01 0.0 103 4.929496E+03 0.0 -1.810325E+04 0.0 2.303275E+03 0.0 110 2.783045E+04 0.0 1.489452E+04 0.0 2.240570E+03 0.0 120 1.341350E+04 0.0 1.311002E+03 0.0 1.048107E+03 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

6-134

-2.230254E+03 -2.306443E+03 -1.380620E+03 1.187097E+04 2.235367E+03 -1.020984E+03 -9.863401E+02 8.316615E+03 3.228048E+03 4.444698E+02 -4.910758E+02 6.037608E+03 2.892530E+03 1.198296E+03 6.942246E+01 4.291236E+03 2.221720E+03 1.289034E+03 4.379200E+02 2.643049E+03 1.390785E+03 9.466555E+02 4.898776E+02

AXIAL TORQUE 0.0 0.0

Static Aeroelastic Analysis Sample Problems

102 -2.910383E-11 0.0 6.192809E+05 103 2.283996E+06 0.0 -1.135454E+07 110 1.722000E+07 0.0 9.448613E+06 120 8.474888E+06 0.0 8.559712E+05 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

0.0 0.0 0.0 0.0

-1.238562E+05 1.363854E+06 1.346043E+06 6.598176E+05

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 SUBCASE

0.0 0.0 3.169128E+06 1.482587E+06 3

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 100 -4.076896E+05 0.0 -5.404446E+05 0.0 2.655099E+04 0.0 0.0 0.0 101 -2.182787E-11 0.0 -2.182787E-11 0.0 0.0 0.0 0.0 0.0 102 -1.455192E-11 0.0 5.188265E+05 0.0 -1.037653E+05 0.0 0.0 0.0 103 -5.404446E+05 0.0 -9.819649E+05 0.0 4.415202E+04 0.0 0.0 0.0 110 1.274054E+06 0.0 9.145390E+05 0.0 6.226979E+04 0.0 0.0 3.983004E+05 120 7.893908E+05 0.0 1.048104E+05 0.0 5.928641E+04 0.0 0.0 1.815371E+05 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 1 S T R E S S E S I N ELEMENT SA1 SA2 SA3 SA4 ID. SB1 SB2 SB3 SB4 100 -9.879942E+04 -9.879942E+04 9.879942E+04 -2.839392E+04 -2.839392E+04 2.839392E+04 101 0.0 0.0 0.0 0.0 0.0 0.0 102 4.092729E-14 4.092729E-14 -4.092729E-14 -1.009861E+03 -1.009861E+03 1.009861E+03 103 -2.839391E+04 -2.839391E+04 2.839391E+04 1.042748E+05 1.042748E+05 -1.042748E+05 110 -8.015175E+04 -8.015175E+04 8.015175E+04 -4.289623E+04 -4.289623E+04 4.289623E+04 120 -3.863090E+04 -3.863090E+04 3.863090E+04 -3.775687E+03 -3.775687E+03 3.775687E+03

B A R E L E M E N T S ( C B A R ) AXIAL SA-MAX SA-MIN M.S.-T STRESS SB-MAX SB-MIN M.S.-C 9.879942E+04 0.0 9.879942E+04 -9.879942E+04 2.839392E+04 2.839392E+04 -2.839392E+04 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.092729E-14 0.0 4.092729E-14 -4.092729E-14 1.009861E+03 1.009861E+03 -1.009861E+03 2.839391E+04 0.0 2.839391E+04 -2.839391E+04 -1.042748E+05 1.042748E+05 -1.042748E+05 8.015175E+04 0.0 8.015175E+04 -8.015175E+04 4.289623E+04 4.289623E+04 -4.289623E+04 3.863090E+04 0.0 3.863090E+04 -3.863090E+04 3.775687E+03 3.775687E+03 -3.775687E+03

HALF-SPAN MODEL, STATIC SYMMETRIC LOADING S T R E S S E S I N ELEMENT SA1 SA2 SA3 SA4 ID. SB1 SB2 SB3 SB4 100 -5.426055E+07 -5.426055E+07 5.426055E+07 -1.315582E+07 -1.315582E+07 1.315582E+07 101 8.381908E-11 8.381908E-11 -8.381908E-11 -1.257286E-10 -1.257286E-10 1.257286E-10 102 1.676382E-10 1.676382E-10 -1.676382E-10 -3.567060E+06 -3.567060E+06 3.567060E+06 103 -1.315582E+07 -1.315582E+07 1.315582E+07 6.540219E+07 6.540219E+07 -6.540219E+07 110 -4.959362E+07 -4.959362E+07 4.959362E+07 -2.721202E+07 -2.721202E+07 2.721202E+07 120 -2.440769E+07 -2.440769E+07 2.440769E+07 -2.465199E+06 -2.465199E+06 2.465199E+06 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING ELEMENT SA1 ID. SB1 100 2.348294E+06 3.112963E+06 101 1.257286E-10 1.257286E-10 102 8.381908E-11 -2.988442E+06 103 3.112963E+06 5.656122E+06 110 -3.669277E+06 -2.633874E+06 120 -2.273447E+06 -3.018542E+05

S T R E SA2 SB2 2.348294E+06 3.112963E+06 1.257286E-10 1.257286E-10 8.381908E-11 -2.988442E+06 3.112963E+06 5.656122E+06 -3.669277E+06 -2.633874E+06 -2.273447E+06 -3.018542E+05

S S E S I N SA3 SB3 -2.348294E+06 -3.112963E+06 -1.257286E-10 -1.257286E-10 -8.381908E-11 2.988442E+06 -3.112963E+06 -5.656122E+06 3.669277E+06 2.633874E+06 2.273447E+06 3.018542E+05

SUBCASE 2 B A R

E L E M E N T S ( C AXIAL SA-MAX STRESS SB-MAX 5.426055E+07 0.0 5.426055E+07 1.315582E+07 1.315582E+07 -8.381908E-11 0.0 8.381908E-11 1.257286E-10 1.257286E-10 -1.676382E-10 0.0 1.676382E-10 3.567060E+06 3.567060E+06 1.315582E+07 0.0 1.315582E+07 -6.540219E+07 6.540219E+07 4.959362E+07 0.0 4.959362E+07 2.721202E+07 2.721202E+07 2.440769E+07 0.0 2.440769E+07 2.465199E+06 2.465199E+06 SUBCASE

B A R ) SA-MIN M.S.-T SB-MIN M.S.-C -5.426055E+07 -1.315582E+07 -8.381908E-11 -1.257286E-10 -1.676382E-10 -3.567060E+06 -1.315582E+07 -6.540219E+07 -4.959362E+07 -2.721202E+07 -2.440769E+07 -2.465199E+06 3

B A R E L E M E N T S ( C B A R ) SA4 AXIAL SA-MAX SA-MIN M.S.-T SB4 STRESS SB-MAX SB-MIN M.S.-C -2.348294E+06 0.0 2.348294E+06 -2.348294E+06 -3.112963E+06 3.112963E+06 -3.112963E+06 -1.257286E-10 0.0 1.257286E-10 -1.257286E-10 -1.257286E-10 1.257286E-10 -1.257286E-10 -8.381908E-11 0.0 8.381908E-11 -8.381908E-11 2.988442E+06 2.988442E+06 -2.988442E+06 -3.112963E+06 0.0 3.112963E+06 -3.112963E+06 -5.656122E+06 5.656122E+06 -5.656122E+06 3.669277E+06 0.0 3.669277E+06 -3.669277E+06 2.633874E+06 2.633874E+06 -2.633874E+06 2.273447E+06 0.0 2.273447E+06 -2.273447E+06 3.018542E+05 3.018542E+05 -3.018542E+05

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Listing 6-28. Output for Camber/Twist Loading of Example HA144GA N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA144GB $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA144GB $$$$$$$$ $ $ $ MODEL DESCRIPTION 30 DEG FWD SWEPT WING W/CANARD $ $ BEAM MODEL WITH DUMBBELL MASSES $ $ $ $ SOLUTION SYMMETRIC IN-FLIGHT STATIC STABILITY $ $ ANALYSIS USING DOUBLET-LATTICE $ $ METHOD AERODYNAMICS AT MACH NO. 0.9 $ $ AND ZONA51 AERO AT MACH NO. 1.3 $ $ $ OUTPUT PLOTS OF THE STICK MODEL AND AERO $ $ GRID, LISTS OF RESTRAINED AND $ $ UNRESTRAINED SYMMETRIC STATIC $ $ STABILITY DERIVATIVES PLUS THE $ $ AERODYNAMIC FORCES AND PRESSURES $ $ PLUS STRESSES AND DEFLECTIONS FOR $ $ 1G LEVEL FLIGHT. $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ CPU TIME IN MINUTES SOL 144 $ STATIC AERO COMPILE AESTATRS SOUIN=MSCSOU NOREF $ $ UNIT SOLUTION ALTER $ ALTER 116,116 $ PARAML ZZX//’TRAILER’/1/S,N,NUXSET//S,N,NOZZX $ MATGEN /UX/1/NUXSET $ ALTER 124,124 $ MPYAD KALX,UX,/KSALX//-1 $ ALTER 128,128 MPYAD DJX,UX,/WJ1 $ ALTER 131 $ MATPRN FFAJ// $ ALTER 134 $ MATPRN PAK// $ ALTER 138,138 ALTER 140 $ JUMP ENDAS $ ALTER 144 $ LABEL ENDAS $ $ $ END OF ALTER $ ENDALTER CEND EXAMPLE HA144GB: 30 DEG FWD SWEPT WING WITH CANARD PAGE 7 STABILITY DERIVATIVES AND UNIT INERTIAL SOLUTIONS HALF-SPAN MODEL, STATIC SYMMETRIC LOADING C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA144GB: 30 DEG FWD SWEPT WING WITH CANARD 2 SUBTI = STABILITY DERIVATIVES AND UNIT INERTIAL SOLUTIONS 3 LABEL = HALF-SPAN MODEL, STATIC SYMMETRIC LOADING 4 ECH0 = BOTH 5 SPC = 1 $ SYMMETRIC CONSTRAINTS 6 DISP = ALL $ PRINT ALL DISPLACEMENTS 7 STRESS = ALL $ PRINT ALL STRESSES 8 FORCE = ALL $ PRINT ALL FORCES 9 AEROF = ALL $ PRINT ALL AERODYNAMIC FORCES 10 APRES = ALL $ PRINT ALL AERODYNAMIC PRESSURES 11 SUBCASE 1 12 TRIM = 1 $ ANGLEA SOLUTION 13 SUBCASE 2 14 TRIM = 1 $ PITCH RATE SOLUTION 15 SUBCASE 3 16 TRIM = 1 $ URDD3 SOLUTION 17 SUBCASE 4 18 TRIM = 1 $ URRD5 SOLUTION 19 SUBCASE5 20 TRIM = 1 $ ELEV SOLUTION

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21

BEGIN BULK

Listing 6-29. Input File for Unit Trim Variable Solutions of Example HA144GB EXAMPLE HA144GB: 30 DEG FWD SWEPT WING WITH CANARD PAGE STABILITY DERIVATIVES AND UNIT INERTIAL SOLUTIONS HALF-SPAN MODEL, STATIC SYMMETRIC LOADING S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 1AELIST 1000 1000 THRU 1007 2AEROS 1 100 10.0 40.0 200.0 1 3AESTAT 501 ANGLEA 4AESTAT 502 PITCH 5AESTAT 503 URDD3 6AESTAT 504 URDD5 7AESURF 505 ELEV 1 1000 8CAERO1 1000 1000 2 4 9+CAC 10. 0. 0. 10. 10. 5. 0. 10CAERO1 1100 1000 8 4 11+CAW 25. 0. 0. 10. 13.45299+20. 0. 12CBAR 100 100 90 99 0. 0. 1. 13CBAR 101 100 97 98 0. 0. 1. 14CBAR 102 100 98 90 0. 0. 1. 15CBAR 103 100 99 100 0. 0. 1. 16CBAR 110 101 100 110 0. 0. 1. 17CBAR 120 101 110 120 0. 0. 1. 18CONM2 97 97 0 1500.0 19CONM2 98 98 0 1500.0 20CONM2 99 99 0 1500.0 21CONM2 100 100 0 1500.0 22CONM2 111 111 0 600.0 23CONM2 112 112 0 400.0 24CONM2 121 121 0 600.0 25CONM2 122 122 0 400.0 26CORD2R 1 0 12.5 0. 0. 12.5 0. 27+CRD1 20. 0. 0. 28CORD2R 2 0 30. 0. 0. 30. 0. 29+CRD2 38.66025+5.0 0. 30CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 31+CRD100 0.0 0.0 0.0 32DMI FA2J 0 2 1 0 40 33DMI FA2J 1 1 0.0 THRU 40 34DMI FA2J 2 1 0.0 THRU 40 35DMI FA2J 3 1 0.0 THRU 40 36DMI W2GJ 0 2 1 0 40 37DMI W2GJ 1 9 .0017453THRU 40 38DMI W2GJ 2 9 .0 THRU 40 39DMI W2GJ 3 9 .0 THRU 40 40DMI WKK 0 3 1 0 80 41DMI WKK 1 1 1.0 THRU 80 42GRID 90 15. 0. 0. 43GRID 97 0. 0. 0. 44GRID 98 10. 0. 0. 45GRID 99 20. 0. 0. 46GRID 100 30. 0. 0. 47GRID 110 27.11325+5. 0. 48GRID 111 24.61325+5. 0. 49GRID 112 29.61325+5. 0. 50GRID 120 21.33975+15. 0. 51GRID 121 18.83975+15. 0. 52GRID 122 23.83975+15. 0. 53MAT1 1 1.44+9 5.40+8 54OMIT1 4 110 120 55PAERO1 1000 56PARAM AUNITS .031081 57PARAM GRDPNT 90 58PARAM WTMASS .031081 59PBAR 100 1 2.0 .173611 0.15 0.5 60+PB1 1.0 1.0 1.0 -1.0 -1.0 1.0 -1.0 61+PB2 0.0 62PBAR 101 1 1.5 0.173611+2.0 0.462963 63+PB3 0.5 3.0 0.5 -3.0 -0.5 3.0 -0.5 64+PB4 0.0 65RBAR 111 110 111 123456 66RBAR 112 110 112 123456 67RBAR 121 120 121 123456

16

8

..

9

..

1 10. 1 10.

+CAC

10.

+CRD1

10

.

+CAW

10.

+CRD2

-10.0

+CRD100

3

3

1

-1.0

+PB1 +PB2

-3.0

+PB3 +PB4

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Chapter 6

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686970717273747576777879-

122 1000 1100 1 1 1501 1. 1601 -1. 90 1 URDD3

RBAR SET1 SET1 SPC1 SPC1 SPLINE2 +SPC SPLINE2 +SPW SUPORT TRIM +TR1

120 122 98 99 99 100 246 97 1246 90 1000 1000 -1. 1100 1100 -1. 35 0.9 1200.0 0.0 URDD5 ENDDATA TOTAL COUNT=

123456 111 98

112 99

121 100

122

1007

1000

0.

1.

1

+SPC

1131

1100

0.

1.

2

+SPW

ANGLEA 0.0

0.0 ELEV

PITCH 0.0

0.0

+TR1

80

Listing 6-30. Sorted Bulk Data for Unit Trim Bariable Solutions of Example HS144GB EXAMPLE HA144GB: 30 DEG FWD SWEPT WING WITH CANARD PAGE 22 STABILITY DERIVATIVES AND UNIT INERTIAL SOLUTIONS HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 1 N O N - D I M E N S I O N A L S T A B I L I T Y A N D C O N T R O L D E R I V A T I V E C O E F F I C I E N T S MACH = 9.0000E-01 Q = 1.2000E+03 TRANSFORMATION FROM BASIC TO REFERENCE COORDINATES: { X } [ -1.0000 0.0000 0.0000 ] { X } { 1.5000E+01 } { Y } = [ 0.0000 1.0000 0.0000 ] { Y } + { 0.0000E+00 } { Z }REF [ 0.0000 0.0000 -1.0000 ] { Z }BAS { 0.0000E+00 } TRIM VARIABLE COEFFICIENT RIGID ELASTIC UNSPLINED SPLINED RESTRAINED UNRESTRAINED INTERCEPT CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -8.420787E-03 -8.420787E-03 -1.033208E-02 -1.265279E-02 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -6.008128E-03 -6.008128E-03 -7.073892E-03 -8.678378E-03 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ANGLEA CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -5.070976E+00 -5.070976E+00 -6.462928E+00 -7.771568E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -2.870931E+00 -2.870932E+00 -3.667061E+00 -4.576804E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 PITCH CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -1.207429E+01 -1.207429E+01 -1.285578E+01 -1.610021E+01 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY -9.953999E+00 -9.953999E+00 -1.027400E+01 -1.249935E+01 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 URDD3 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 3.634422E-03 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 2.623634E-03 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 URDD5 CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ 0.000000E+00 0.000000E+00 7.244892E-02 0.000000E+00 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 0.000000E+00 0.000000E+00 4.702159E-02 0.000000E+00 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 ELEV CX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CY 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CZ -2.461395E-01 -2.461395E-01 -5.429845E-01 -5.219331E-01 CMX 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 CMY 5.715299E-01 5.715299E-01 3.860484E-01 3.956231E-01 CMZ 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 INTERMEDIATE MATRIX ... HP0 COLUMN 1 2.373158E-03 -1.418578E-04 INTERMEDIATE MATRIX ... HP COLUMN 1 1.414911E+00 -7.485947E-02 COLUMN 2 2.982306E+00 -2.136336E-01 COLUMN 3 5.546751E-03 -1.107548E-04 COLUMN 4 -1.318791E-03 -7.283140E-03 COLUMN 5 5.516857E-02 6.420408E-03 MATRIX FFAJ (GINO NAME 101 ) IS A DB PREC 5 COLUMN X 40 ROW RECTANG MATRIX. COLUMN 1 ROWS 1 THRU 40 -------------------------------------------------ROW 1) 9.1666D+03 2.1210D+03 1.5483D+03 1.9278D+03 7.2503D+03 1.5584D+03 1.2235D+03 1.7198D+03 1.5493D+04 11) 1.6704D+03 5.9146D+02 1.6143D+04 5.6137D+03 2.4273D+03 6.7285D+02 2.5019D+04 7.4582D+03 3.2205D+03 21) 2.1220D+04 8.5260D+03 4.1467D+03 1.7152D+03 1.8212D+04 8.1171D+03 4.5263D+03 2.1512D+03 1.5442D+04 31) 4.3658D+03 2.3120D+03 1.2408D+04 5.6879D+03 3.7397D+03 2.2080D+03 8.4571D+03 3.7568D+03 2.5752D+03 COLUMN 2 ROWS 1 THRU 40 -------------------------------------------------ROW 1 -3.2985D+03 4.4863D+03 6.2786D+03 7.0460D+03 -2.7056D+03 3.5977D+03 4.9634D+03 5.8645D+03 3.9753D+04 11) 1.0228D+04 5.8001D+03 3.3939D+04 1.8425D+04 1.1910D+04 6.6435D+03 3.7854D+04 1.9368D+04 1.3071D+04 21) 2.9675D+04 1.8714D+04 1.3760D+04 8.5518D+03 2.2736D+04 1.6467D+04 1.3327D+04 8.9174D+03 1.6563D+04 31) 1.2051D+04 8.6812D+03 1.0893D+04 1.0288D+04 9.9850D+03 7.8852D+03 5.6418D+03 6.4447D+03 6.7800D+03

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COLUMN 3 ROWS 1 THRU 40 -------------------------------------------------ROW 1)-9.1347D-01 -4.6297D-01 -6.3842D-01 -9.7517D-01 -7.1823D-01 -3.5457D-01 -5.1301D-01 -8.6670D-01 -1.4105D+01 11)-1.8447D+00 -7.0096D-01 -1.1642D+01 -5.3030D+00 -2.4882D+00 -1.0801D+00 -1.1920D+01 -5.1602D+00 -2.9185D+00 21)-1.0108D+01 -4.8003D+00 -2.9487D+00 -1.5464D+00 -8.6765D+00 -4.2041D+00 -2.7500D+00 -1.5631D+00 -7.3460D+00 31)-2.4086D+00 -1.4593D+00 -5.9000D+00 -2.7821D+00 -1.9532D+00 -1.2748D+00 -4.0242D+00 -1.8164D+00 -1.3012D+00 COLUMN 4 ROWS 1 THRU 40 -------------------------------------------------ROW 1)-4.1293D+01 -1.3475D+01 -1.4490D+01 -2.0731D+01 -3.2596D+01 -1.0132D+01 -1.1604D+01 -1.8516D+01 -2.3530D+02 11)-2.4670D+01 -8.7756D+00 -2.1791D+02 -8.0090D+01 -3.5193D+01 -1.1034D+01 -2.6029D+02 -9.4343D+01 -4.4381D+01 21)-2.2531D+02 -9.7074D+01 -5.2439D+01 -2.3477D+01 -1.9607D+02 -8.9374D+01 -5.3335D+01 -2.7302D+01 -1.6767D+02 31)-4.9371D+01 -2.7626D+01 -1.3552D+02 -6.2051D+01 -4.1522D+01 -2.5468D+01 -9.2801D+01 -4.1009D+01 -2.8331D+01 COLUMN 5 ROWS 1 THRU 40 -------------------------------------------------ROW 1) 8.5722D+03 1.4786D+03 4.8162D+02 2.5855D+02 6.7884D+03 1.0565D+03 3.6013D+02 2.2569D+02 -6.0992D+03 11)-4.6103D+02 -1.5862D+02 -3.1653D+03 -1.5109D+03 -6.1258D+02 -2.5892D+02 4.5269D+03 -5.7095D+02 -5.9045D+02 21) 3.0390D+03 5.7225D+02 -2.6137D+02 -2.5250D+02 2.1291D+03 8.2638D+02 1.1378D+02 -1.2572D+02 1.5456D+03 31) 3.0676D+02 1.7772D+01 1.0986D+03 5.6876D+02 3.2999D+02 1.1211D+02 6.7662D+02 3.5604D+02 2.4234D+02 THE NUMBER OF NON-ZERO TERMS IN THE DENSEST COLUMN = 40 THE DENSITY OF THIS MATRIX IS 100.00 PERCENT. MATRIX PAK (GINO NAME 101 ) IS A DB PREC 5 COLUMN X 80 ROW RECTANG MATRIX. COLUMN 1 ROWS 1 THRU 80 -------------------------------------------------ROW 1) 5.7291D+04 3.5807D+04 1.3256D+04 8.2851D+03 9.6766D+03 6.0479D+03 1.2049D+04 7.5304D+03 4.5315D+04 11) 9.7400D+03 6.0875D+03 7.6468D+03 4.7793D+03 1.0749D+04 6.7180D+03 9.6831D+04 6.0519D+04 2.7211D+04 21) 1.0440D+04 6.5251D+03 3.6966D+03 2.3104D+03 1.0089D+05 6.3059D+04 3.5085D+04 2.1928D+04 1.5170D+04 31) 4.2053D+03 2.6283D+03 1.5637D+05 9.7730D+04 4.6614D+04 2.9134D+04 2.0128D+04 1.2580D+04 3.9282D+03 41) 1.3263D+05 8.2892D+04 5.3287D+04 3.3305D+04 2.5917D+04 1.6198D+04 1.0720D+04 6.7000D+03 1.1383D+05 51) 5.0732D+04 3.1708D+04 2.8290D+04 1.7681D+04 1.3445D+04 8.4030D+03 9.6514D+04 6.0321D+04 4.4281D+04 61) 2.7286D+04 1.7054D+04 1.4450D+04 9.0314D+03 7.7551D+04 4.8469D+04 3.5550D+04 2.2218D+04 2.3373D+04 71) 1.3800D+04 8.6250D+03 5.2857D+04 3.3036D+04 2.3480D+04 1.4675D+04 1.6095D+04 1.0059D+04 1.0727D+04 COLUMN 2 ROWS 1 THRU 80 -------------------------------------------------ROW 1) -2.0616D+04 -1.2885D+04 2.8040D+04 1.7525D+04 3.9241D+04 2.4526D+04 4.4038D+04 2.7523D+04 -1.6910D+04 11) 2.2486D+04 1.4054D+04 3.1021D+04 1.9388D+04 3.6653D+04 2.2908D+04 2.4846D+05 1.5529D+05 1.0288D+05 21) 6.3922D+04 3.9951D+04 3.6251D+04 2.2657D+04 2.1212D+05 1.3257D+05 1.1515D+05 7.1971D+04 7.4437D+04 31) 4.1522D+04 2.5951D+04 2.3659D+05 1.4787D+05 1.2105D+05 7.5655D+04 8.1694D+04 5.1059D+04 4.3465D+04 41) 1.8547D+05 1.1592D+05 1.1696D+05 7.3102D+04 8.6001D+04 5.3751D+04 5.3449D+04 3.3405D+04 1.4210D+05 51) 1.0292D+05 6.4324D+04 8.3294D+04 5.2059D+04 5.5734D+04 3.4834D+04 1.0352D+05 6.4698D+04 8.4792D+04 61) 7.5316D+04 4.7072D+04 5.4258D+04 3.3911D+04 6.8084D+04 4.2552D+04 6.4300D+04 4.0187D+04 6.2406D+04 71) 4.9282D+04 3.0801D+04 3.5261D+04 2.2038D+04 4.0279D+04 2.5175D+04 4.2375D+04 2.6484D+04 3.8093D+04 COLUMN 3 ROWS 1 THRU 80 -------------------------------------------------ROW 1) -5.7092D+00 -3.5682D+00 -2.8936D+00 -1.8085D+00 -3.9901D+00 -2.4938D+00 -6.0948D+00 -3.8092D+00 -4.4889D+00 11) -2.2161D+00 -1.3851D+00 -3.2063D+00 -2.0040D+00 -5.4169D+00 -3.3855D+00 -8.8156D+01 -5.5097D+01 -2.6338D+01 21) -1.1530D+01 -7.2060D+00 -4.3810D+00 -2.7381D+00 -7.2760D+01 -4.5475D+01 -3.3144D+01 -2.0715D+01 -1.5551D+01 31) -6.7509D+00 -4.2193D+00 -7.4502D+01 -4.6564D+01 -3.2251D+01 -2.0157D+01 -1.8240D+01 -1.1400D+01 -8.7912D+00 41) -6.3175D+01 -3.9485D+01 -3.0002D+01 -1.8751D+01 -1.8429D+01 -1.1518D+01 -9.6650D+00 -6.0406D+00 -5.4228D+01 51) -2.6276D+01 -1.6422D+01 -1.7187D+01 -1.0742D+01 -9.7696D+00 -6.1060D+00 -4.5913D+01 -2.8695D+01 -2.2140D+01 61) -1.5054D+01 -9.4085D+00 -9.1205D+00 -5.7003D+00 -3.6875D+01 -2.3047D+01 -1.7388D+01 -1.0868D+01 -1.2207D+01 71) -7.9674D+00 -4.9796D+00 -2.5151D+01 -1.5720D+01 -1.1353D+01 -7.0955D+00 -8.1325D+00 -5.0828D+00 -5.8166D+00 COLUMN 4 ROWS 1 THRU 80 -------------------------------------------------ROW 1) -2.5808D+02 -1.6130D+02 -8.4219D+01 -5.2637D+01 -9.0560D+01 -5.6600D+01 -1.2957D+02 -8.0979D+01 -2.0373D+02 11) -6.3323D+01 -3.9577D+01 -7.2524D+01 -4.5328D+01 -1.1573D+02 -7.2330D+01 -1.4706D+03 -9.1913D+02 -3.9720D+02 21) -1.5419D+02 -9.6368D+01 -5.4847D+01 -3.4280D+01 -1.3620D+03 -8.5122D+02 -5.0056D+02 -3.1285D+02 -2.1996D+02 31) -6.8960D+01 -4.3100D+01 -1.6268D+03 -1.0168D+03 -5.8965D+02 -3.6853D+02 -2.7738D+02 -1.7336D+02 -6.9528D+01 41) -1.4082D+03 -8.8013D+02 -6.0671D+02 -3.7919D+02 -3.2775D+02 -2.0484D+02 -1.4673D+02 -9.1708D+01 -1.2254D+03 51) -5.5859D+02 -3.4912D+02 -3.3334D+02 -2.0834D+02 -1.7063D+02 -1.0665D+02 -1.0480D+03 -6.5497D+02 -4.8410D+02 61) -3.0857D+02 -1.9286D+02 -1.7266D+02 -1.0791D+02 -8.4702D+02 -5.2939D+02 -3.8782D+02 -2.4239D+02 -2.5951D+02 71) -1.5917D+02 -9.9484D+01 -5.8000D+02 -3.6250D+02 -2.5631D+02 -1.6019D+02 -1.7707D+02 -1.1067D+02 -1.2078D+02 COLUMN 5 ROWS 1 THRU 80 -------------------------------------------------ROW 1) 5.3576D+04 3.3485D+04 9.2415D+03 5.7759D+03 3.0101D+03 1.8813D+03 1.6159D+03 1.0100D+03 4.2427D+04 11) 6.6029D+03 4.1268D+03 2.2508D+03 1.4068D+03 1.4106D+03 8.8160D+02 -3.8120D+04 -2.3825D+04 -7.9864D+03 21)-2.8814D+03 -1.8009D+03 -9.9138D+02 -6.1961D+02 -1.9783D+04 -1.2364D+04 -9.4432D+03 -5.9020D+03 -3.8286D+03 31)-1.6183D+03 -1.0114D+03 2.8293D+04 1.7683D+04 -3.5684D+03 -2.2303D+03 -3.6903D+03 -2.3064D+03 -2.2090D+03 41) 1.8994D+04 1.1871D+04 3.5766D+03 2.2354D+03 -1.6336D+03 -1.0210D+03 -1.5781D+03 -9.8634D+02 1.3307D+04 51) 5.1649D+03 3.2280D+03 7.1115D+02 4.4447D+02 -7.8572D+02 -4.9108D+02 9.6602D+03 6.0376D+03 4.6280D+03 61) 1.9173D+03 1.1983D+03 1.1108D+02 6.9422D+01 6.8660D+03 4.2912D+03 3.5548D+03 2.2217D+03 2.0625D+03 71) 7.0067D+02 4.3792D+02 4.2289D+03 2.6430D+03 2.2253D+03 1.3908D+03 1.5146D+03 9.4666D+02 7.8380D+02 80 THE NUMBER OF NON-ZERO TERMS IN THE DENSEST COLUMN = THE DENSITY OF THIS MATRIX IS 100.00 PERCENT. T A B L E A E R O S T A T I C D A T A R E C O V E R Y O U T P U T AEROELASTIC TRIM VARIABLES TRIM VARIABLE VALUE OF UX ANGLEA 1.000000E+00 PITCH 0.000000E+00 URDD3 0.000000E+00 URDD5 0.000000E+00 ELEV 0.000000E+00 SUBCASE 1 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING A E R O S T A T I C D A T A R E C O V E R Y O U T P U T T A B L E MACH = 9.000000E-01 Q = 1.200000E+03 AERODYNAMIC FORCES AERODYNAMIC AERODYNAMIC AERODYNAMIC PRES. GRID LABEL COEFFICIENTS PRESSURES EXTERNAL ID LABEL NORMAL FORCES(T3) MOMENTS(R2) 1000 LS 5.729107E+04 3.580691E+04 1 LS 7.638809E+00 9.166570E+03 2 LS 1.767498E+00 2.120998E+03 1001 LS 1.325624E+04 8.285148E+03 1002 LS 9.676568E+03 6.047855E+03 3 LS 1.290209E+00 1.548251E+03 4 LS 1.606494E+00 1.927792E+03 1003 LS 1.204870E+04 7.530439E+03

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5 LS 6.041941E+00 7.250330E+03 1004 LS 4.531456E+04 6 LS 1.298661E+00 1.558393E+03 1005 LS 9.739958E+03 7 LS 1.019578E+00 1.223494E+03 1006 LS 7.646835E+03 8 LS 1.433167E+00 1.719800E+03 1007 LS 1.074875E+04 9 LS 1.291075E+01 1.549290E+04 1100 LS 9.683061E+04 10 LS 3.628104E+00 4.353725E+03 1101 LS 2.721078E+04 11 LS 1.392011E+00 1.670413E+03 1102 LS 1.044008E+04 12 LS 4.928862E-01 5.914635E+02 1103 LS 3.696647E+03 13 LS 1.345262E+01 1.614314E+04 1104 LS 1.008946E+05 14 LS 4.678058E+00 5.613669E+03 1105 LS 3.508543E+04 15 LS 2.022729E+00 2.427275E+03 1106 LS 1.517047E+04 16 LS 5.607044E-01 6.728453E+02 1107 LS 4.205283E+03 17 LS 2.084907E+01 2.501889E+04 1108 LS 1.563680E+05 18 LS 6.215158E+00 7.458190E+03 1109 LS 4.661369E+04 19 LS 2.683741E+00 3.220489E+03 1110 LS 2.012806E+04 20 LS 5.237615E-01 6.285137E+02 1111 LS 3.928211E+03 21 LS 1.768373E+01 2.122048E+04 1112 LS 1.326280E+05 22 LS 7.104998E+00 8.525997E+03 1113 LS 5.328748E+04 23 LS 3.455575E+00 4.146690E+03 1114 LS 2.591681E+04 24 LS 1.429336E+00 1.715203E+03 1115 LS 1.072002E+04 25 LS 1.517674E+01 1.821209E+04 1116 LS 1.138256E+05 26 LS 6.764285E+00 8.117142E+03 1117 LS 5.073214E+04 27 LS 3.771950E+00 4.526340E+03 1118 LS 2.828963E+04 28 LS 1.792641E+00 2.151169E+03 1119 LS 1.344481E+04 29 LS 1.286854E+01 1.544225E+04 1120 LS 9.651405E+04 30 LS 5.904088E+00 7.084905E+03 1121 LS 4.428066E+04 31 LS 3.638149E+00 4.365779E+03 1122 LS 2.728612E+04 32 LS 1.926708E+00 2.312049E+03 1123 LS 1.445031E+04 33 LS 1.034013E+01 1.240815E+04 1124 LS 7.755095E+04 34 LS 4.739940E+00 5.687928E+03 1125 LS 3.554955E+04 35 LS 3.116378E+00 3.739654E+03 1126 LS 2.337284E+04 36 LS 1.840009E+00 2.208011E+03 1127 LS 1.380007E+04 37 LS 7.047599E+00 8.457119E+03 1128 LS 5.285699E+04 38 LS 3.130690E+00 3.756828E+03 1129 LS 2.348017E+04 39 LS 2.145972E+00 2.575166E+03 1130 LS 1.609479E+04 40 LS 1.430286E+00 1.716344E+03 1131 LS 1.072715E+04 *** LABEL NOTATIONS: LS = LIFTING SURFACE, ZIB = Z INTERFERENCE BODY ELEMENT, ZSB = Z SLENDER BODY ELEMENT, YIB = Y INTERFERENCE BODY ELEMENT, YSB = Y SLENDER BODY ELEMENT. HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 1 D I S P L A C E M E N T POINT ID. TYPE T1 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T2

T3 0.0 8.257084E-02 2.064271E-02 3.520744E-01 1.070061E+00 2.231179E+00 2.935486E+00 1.526873E+00 7.741362E+00 8.863285E+00 6.619438E+00

D I S P L A C E M E N T POINT ID. TYPE T1 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T2

T3 0.0 3.028352E-02 7.570880E-03 9.909063E-01 4.510385E+00 5.711089E+00 6.123365E+00 5.298812E+00 1.329469E+01 1.424489E+01 1.234449E+01

D I S P L A C E M E N T POINT ID. TYPE T1 90 G 0.0 0.0 97 G 0.0 0.0 98 G 0.0 0.0 99 G 0.0 0.0 100 G 0.0 0.0 110 G 0.0 0.0 111 G 0.0 0.0 112 G 0.0 0.0 120 G 0.0 0.0 121 G 0.0 0.0 122 G 0.0 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T2

T3 0.0 7.737751E-03 1.246937E-03 1.707710E-03 1.011274E-02 8.358140E-03 6.424596E-03 1.029169E-02 7.476539E-03 5.733185E-03 9.219893E-03

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2.832160E+04 6.087474E+03 4.779272E+03 6.717970E+03 6.051913E+04 1.700674E+04 6.525054E+03 2.310404E+03 6.305914E+04 2.192839E+04 9.481546E+03 2.628302E+03 9.773002E+04 2.913355E+04 1.258004E+04 2.455132E+03 8.289248E+04 3.330468E+04 1.619801E+04 6.700013E+03 7.114098E+04 3.170758E+04 1.768102E+04 8.403006E+03 6.032128E+04 2.767541E+04 1.705383E+04 9.031441E+03 4.846934E+04 2.221847E+04 1.460802E+04 8.625043E+03 3.303562E+04 1.467511E+04 1.005924E+04 6.704467E+03

V E C T O R R1 R2 0.0 0.0 0.0 6.192812E-03 0.0 6.192812E-03 0.0 -1.170423E-01 0.0 6.436867E-02 2.300923E-01 2.817225E-01 2.300923E-01 2.817225E-01 2.300923E-01 2.817225E-01 3.824708E-01 4.487695E-01 3.824708E-01 4.487695E-01 3.824708E-01 4.487695E-01 SUBCASE 2

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

V E C T O R R1 R2 0.0 0.0 0.0 2.271264E-03 0.0 2.271264E-03 0.0 -3.456965E-01 0.0 -1.707524E-01 4.143655E-01 1.649105E-01 4.143655E-01 1.649105E-01 4.143655E-01 1.649105E-01 6.728314E-01 3.800796E-01 6.728314E-01 3.800796E-01 6.728314E-01 3.800796E-01 SUBCASE 3

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

V E C T O R R1 R2 0.0 0.0 0.0 7.490814E-04 0.0 4.490812E-04 0.0 -6.139819E-04 0.0 -8.868701E-04 2.251395E-04 -7.734181E-04 2.251395E-04 -7.734181E-04 2.251395E-04 -7.734181E-04 3.792083E-04 -6.973416E-04 3.792083E-04 -6.973416E-04 3.792083E-04 -6.973416E-04 SUBCASE 4 V E C T O R

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Static Aeroelastic Analysis Sample Problems

POINT ID. TYPE T1 90 G 0.0 97 G 0.0 98 G 0.0 99 G 0.0 100 G 0.0 110 G 0.0 111 G 0.0 112 G 0.0 120 G 0.0 121 G 0.0 122 G 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T3 0.0 -1.066846E-01 -1.635863E-02 1.943326E-02 1.332182E-01 9.112068E-02 5.363703E-02 1.286043E-01 -3.389182E-03 -4.381344E-02 3.703507E-02

D I S P L A C E M E N T POINT ID. TYPE T1 90 G 0.0 97 G 0.0 98 G 0.0 99 G 0.0 100 G 0.0 110 G 0.0 111 G 0.0 112 G 0.0 120 G 0.0 121 G 0.0 122 G 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R1 R2 0.0 0.0 0.0 -1.053259E-02 0.0 -6.032590E-03 0.0 -7.261352E-03 0.0 -1.388727E-02 -2.158867E-04 -1.499346E-02 -2.158867E-04 -1.499346E-02 -2.158867E-04 -1.499346E-02 -5.469111E-04 -1.616970E-02 -5.469111E-04 -1.616970E-02 -5.469111E-04 -1.616970E-02 SUBCASE 5

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

V E C T O R

T3

R1 0.0 0.0 0.0 0.0 0.0 1.728677E-02 1.728677E-02 1.728677E-02 3.097839E-02 3.097839E-02 3.097839E-02 SUBCASE

0.0 6.917690E-02 1.729423E-02 -2.259708E-02 -2.549343E-01 -6.271992E-02 8.860861E-02 -2.140484E-01 6.092786E-01 8.045742E-01 4.139830E-01

F O R C E S I N B A R E L E M E N T S ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 100 9.420228E+06 0.0 2.283996E+06 0.0 1.427247E+06 0.0 101 -1.455192E-11 0.0 2.182787E-11 0.0 -3.637979E-12 0.0 102 -2.910383E-11 0.0 6.192809E+05 0.0 -1.238562E+05 0.0 103 2.283996E+06 0.0 -1.135454E+07 0.0 1.363854E+06 0.0 110 1.722000E+07 0.0 9.448613E+06 0.0 1.346043E+06 0.0 120 8.474888E+06 0.0 8.559712E+05 0.0 6.598176E+05 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING F O R C E S I N B A R E L E M E N ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 100 2.488472E+07 0.0 9.684911E+06 0.0 3.039961E+06 101 -3.637979E-12 0.0 -3.637979E-12 0.0 0.0 102 -1.455192E-11 0.0 2.271263E+05 0.0 -4.542525E+04 103 9.684910E+06 0.0 -1.843211E+07 0.0 2.811702E+06 110 3.060099E+07 0.0 1.501103E+07 0.0 2.700260E+06 120 1.363715E+07 0.0 7.138633E+05 0.0 1.119189E+06 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

PLANE 2 0.0 0.0 0.0 0.0 0.0 0.0

F O R C E S I N B A R E L E M E N ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 100 4.106438E+04 0.0 2.033378E+04 0.0 4.146119E+03 101 -9.094947E-13 0.0 1.500000E+04 0.0 -1.500000E+03 102 1.500000E+04 0.0 2.990810E+04 0.0 -2.981619E+03 103 2.033378E+04 0.0 -6.689378E+03 0.0 2.702316E+03 110 1.445256E+04 0.0 7.345504E+03 0.0 1.230979E+03 120 7.564060E+03 0.0 -1.393779E+02 0.0 6.671373E+02 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

PLANE 2 0.0 0.0 0.0 0.0 0.0 0.0

R2 0.0 5.188268E-03 5.188268E-03 9.481348E-03 3.992956E-02 6.053141E-02 6.053141E-02 6.053141E-02 7.811824E-02 7.811824E-02 7.811824E-02

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

( C B A R ) - SHEAR AXIAL FORCE TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.169128E+06 0.0 1.482587E+06 SUBCASE 2

T S

T S

( C B A R ) - SHEAR AXIAL FORCE TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.616088E+06 0.0 1.236449E+06 SUBCASE 3 ( C B A R ) - SHEAR AXIAL FORCE TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -6.199578E+02 0.0 -2.414098E+02 SUBCASE 4

F O R C E S I N B A R E L E M E N T S ( C B A R ) ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B - SHEAR AXIAL ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 PLANE 2 FORCE TORQUE 100 4.398598E+05 0.0 2.862749E+05 0.0 3.071698E+04 0.0 0.0 0.0 101 1.455192E-11 0.0 -2.250000E+05 0.0 2.250000E+04 0.0 0.0 0.0 102 -2.250000E+05 0.0 -3.782586E+05 0.0 3.065172E+04 0.0 0.0 0.0 103 2.862749E+05 0.0 4.502091E+04 0.0 2.412540E+04 0.0 0.0 0.0 110 -2.628852E+04 0.0 -3.780227E+04 0.0 1.994240E+03 0.0 0.0 -3.680798E+04 120 -2.721547E+04 0.0 -1.066428E+04 0.0 -1.433375E+03 0.0 0.0 -1.847108E+04 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING SUBCASE 5 F O R C E S I N B A R E L E M E N T S ELEMENT BEND-MOMENT END-A BEND-MOMENT END-B ID. PLANE 1 PLANE 2 PLANE 1 PLANE 2 PLANE 1 100 -4.076896E+05 0.0 -5.404446E+05 0.0 2.655099E+04 0.0 101 -2.182787E-11 0.0 -2.182787E-11 0.0 0.0 0.0 102 -1.455192E-11 0.0 5.188265E+05 0.0 -1.037653E+05 0.0 103 -5.404446E+05 0.0 -9.819649E+05 0.0 4.415202E+04 0.0 110 1.274054E+06 0.0 9.145390E+05 0.0 6.226979E+04 0.0 120 7.893908E+05 0.0 1.048104E+05 0.0 5.928641E+04 0.0 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING ELEMENT ID. 100

SA1 SB1 -5.426055E+07 -1.315582E+07

( C B A R ) - SHEAR AXIAL PLANE 2 FORCE TORQUE 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.983004E+05 0.0 1.815371E+05 SUBCASE 1

S T R E S S E S I N B A R E L E M E N T S SA2 SA3 SA4 AXIAL SB2 SB3 SB4 STRESS -5.426055E+07 5.426055E+07 5.426055E+07 0.0 -1.315582E+07 1.315582E+07 1.315582E+07

( C B A R ) SA-MAX SA-MIN M.S.-T SB-MAX SB-MIN M.S.-C 5.426055E+07 -5.426055E+07 1.315582E+07 -1.315582E+07

NX Nastran Aeroelastic Analysis User’s Guide

6-141

Chapter 6

Static Aeroelastic Analysis Sample Problems

101

8.381908E-11 8.381908E-11 -8.381908E-11 -1.257286E-10 -1.257286E-10 1.257286E-10 1.676382E-10 1.676382E-10 -1.676382E-10 -3.567060E+06 -3.567060E+06 3.567060E+06 103 -1.315582E+07 -1.315582E+07 1.315582E+07 6.540219E+07 6.540219E+07 -6.540219E+07 110 -4.959362E+07 -4.959362E+07 4.959362E+07 -2.721202E+07 -2.721202E+07 2.721202E+07 120 -2.440769E+07 -2.440769E+07 2.440769E+07 -2.465199E+06 -2.465199E+06 2.465199E+06 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

-8.381908E-11 1.257286E-10 -1.676382E-10 3.567060E+06 1.315582E+07 -6.540219E+07 4.959362E+07 2.721202E+07 2.440769E+07 2.465199E+06

S T R E S S E S I N SA1 SA2 SA3 SB1 SB2 SB3 -1.433360E+08 -1.433360E+08 1.433360E+08 -5.578512E+07 -5.578512E+07 5.578512E+07 101 2.095477E-11 2.095477E-11 -2.095477E-11 2.095477E-11 2.095477E-11 -2.095477E-11 102 8.381908E-11 8.381908E-11 -8.381908E-11 -1.308248E+06 -1.308248E+06 1.308248E+06 103 -5.578512E+07 -5.578512E+07 5.578512E+07 1.061690E+08 1.061690E+08 -1.061690E+08 110 -8.813090E+07 -8.813090E+07 8.813090E+07 -4.323180E+07 -4.323180E+07 4.323180E+07 120 -3.927501E+07 -3.927501E+07 3.927501E+07 -2.055928E+06 -2.055928E+06 2.055928E+06 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

B A R

E L E M E N T S ( C B A R ) SA4 AXIAL SA-MAX SA-MIN M.S.-T SB4 STRESS SB-MAX SB-MIN M.S.-C 1.433360E+08 0.0 1.433360E+08 -1.433360E+08 5.578512E+07 5.578512E+07 -5.578512E+07 -2.095477E-11 0.0 2.095477E-11 -2.095477E-11 -2.095477E-11 2.095477E-11 -2.095477E-11 -8.381908E-11 0.0 8.381908E-11 -8.381908E-11 1.308248E+06 1.308248E+06 -1.308248E+06 5.578512E+07 0.0 5.578512E+07 -5.578512E+07 -1.061690E+08 1.061690E+08 -1.061690E+08 8.813090E+07 0.0 8.813090E+07 -8.813090E+07 4.323180E+07 4.323180E+07 -4.323180E+07 3.927501E+07 0.0 3.927501E+07 -3.927501E+07 2.055928E+06 2.055928E+06 -2.055928E+06 SUBCASE 3

S T R E S S E S I N B A SA1 SA2 SA3 SB1 SB2 SB3 -2.365310E+05 -2.365310E+05 2.365310E+05 -1.171226E+05 -1.171226E+05 1.171226E+05 101 5.238693E-12 5.238693E-12 -5.238693E-12 -8.640005E+04 -8.640005E+04 8.640005E+04 102 -8.640005E+04 -8.640005E+04 8.640005E+04 -1.722708E+05 -1.722708E+05 1.722708E+05 103 -1.171226E+05 -1.171226E+05 1.171226E+05 3.853084E+04 3.853084E+04 -3.853084E+04 110 -4.162341E+04 -4.162341E+04 4.162341E+04 -2.115506E+04 -2.115506E+04 2.115506E+04 120 -2.178451E+04 -2.178451E+04 2.178451E+04 4.014085E+02 4.014085E+02 -4.014085E+02 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

R

E L E M E N T S SA4 AXIAL SB4 STRESS 2.365310E+05 0.0 1.171226E+05 -5.238693E-12 0.0 8.640005E+04 8.640005E+04 0.0 1.722708E+05 1.171226E+05 0.0 -3.853084E+04 4.162341E+04 0.0 2.115506E+04 2.178451E+04 0.0 -4.014085E+02

S T R E S S E S I N SA1 SA2 SA3 SB1 SB2 SB3 -2.533594E+06 -2.533594E+06 2.533594E+06 -1.648945E+06 -1.648945E+06 1.648945E+06 101 -8.381908E-11 -8.381908E-11 8.381908E-11 1.296001E+06 1.296001E+06 -1.296001E+06 102 1.296001E+06 1.296001E+06 -1.296001E+06 2.178771E+06 2.178771E+06 -2.178771E+06 103 -1.648945E+06 -1.648945E+06 1.648945E+06 -2.593206E+05 -2.593206E+05 2.593206E+05 110 7.571099E+04 7.571099E+04 -7.571099E+04 1.088706E+05 1.088706E+05 -1.088706E+05 120 7.838059E+04 7.838059E+04 -7.838059E+04 3.071313E+04 3.071313E+04 -3.071313E+04 HALF-SPAN MODEL, STATIC SYMMETRIC LOADING

B A R

102

ELEMENT ID. 100

ELEMENT ID. 100

ELEMENT ID. 100

ELEMENT ID. 100 101 102 103 110 120

6-142

SA1 SB1 2.348294E+06 3.112963E+06 1.257286E-10 1.257286E-10 8.381908E-11 -2.988442E+06 3.112963E+06 5.656122E+06 -3.669277E+06 -2.633874E+06 -2.273447E+06 -3.018542E+05

S T R E S S E S I N SA2 SA3 SB2 SB3 2.348294E+06 -2.348294E+06 3.112963E+06 -3.112963E+06 1.257286E-10 -1.257286E-10 1.257286E-10 -1.257286E-10 8.381908E-11 -8.381908E-11 -2.988442E+06 2.988442E+06 3.112963E+06 -3.112963E+06 5.656122E+06 -5.656122E+06 -3.669277E+06 3.669277E+06 -2.633874E+06 2.633874E+06 -2.273447E+06 2.273447E+06 -3.018542E+05 3.018542E+05

NX Nastran Aeroelastic Analysis User’s Guide

0.0

8.381908E-11 1.257286E-10 1.676382E-10 3.567060E+06 0.0 1.315582E+07 6.540219E+07 0.0 4.959362E+07 2.721202E+07 0.0 2.440769E+07 2.465199E+06 SUBCASE 2

0.0

-8.381908E-11 -1.257286E-10 -1.676382E-10 -3.567060E+06 -1.315582E+07 -6.540219E+07 -4.959362E+07 -2.721202E+07 -2.440769E+07 -2.465199E+06

( C B A R ) SA-MAX SA-MIN M.S.-T SB-MAX SB-MIN M.S.-C 2.365310E+05 -2.365310E+05 1.171226E+05 -1.171226E+05 5.238693E-12 -5.238693E-12 8.640005E+04 -8.640005E+04 8.640005E+04 -8.640005E+04 1.722708E+05 -1.722708E+05 1.171226E+05 -1.171226E+05 3.853084E+04 -3.853084E+04 4.162341E+04 -4.162341E+04 2.115506E+04 -2.115506E+04 2.178451E+04 -2.178451E+04 4.014085E+02 -4.014085E+02 SUBCASE 4

E L E M E N T S ( C B A R ) SA4 AXIAL SA-MAX SA-MIN M.S.-T SB4 STRESS SB-MAX SB-MIN M.S.-C 2.533594E+06 0.0 2.533594E+06 -2.533594E+06 1.648945E+06 1.648945E+06 -1.648945E+06 8.381908E-11 0.0 8.381908E-11 -8.381908E-11 -1.296001E+06 1.296001E+06 -1.296001E+06 -1.296001E+06 0.0 1.296001E+06 -1.296001E+06 -2.178771E+06 2.178771E+06 -2.178771E+06 1.648945E+06 0.0 1.648945E+06 -1.648945E+06 2.593206E+05 2.593206E+05 -2.593206E+05 -7.571099E+04 0.0 7.571099E+04 -7.571099E+04 -1.088706E+05 1.088706E+05 -1.088706E+05 -7.838059E+04 0.0 7.838059E+04 -7.838059E+04 -3.071313E+04 3.071313E+04 -3.071313E+04 SUBCASE 5 B A R

E L E M E N T S ( C B A R ) SA4 AXIAL SA-MAX SA-MIN M.S.-T SB4 STRESS SB-MAX SB-MIN M.S.-C -2.348294E+06 0.0 2.348294E+06 -2.348294E+06 -3.112963E+06 3.112963E+06 -3.112963E+06 -1.257286E-10 0.0 1.257286E-10 -1.257286E-10 -1.257286E-10 1.257286E-10 -1.257286E-10 -8.381908E-11 0.0 8.381908E-11 -8.381908E-11 2.988442E+06 2.988442E+06 -2.988442E+06 -3.112963E+06 0.0 3.112963E+06 -3.112963E+06 -5.656122E+06 5.656122E+06 -5.656122E+06 3.669277E+06 0.0 3.669277E+06 -3.669277E+06 2.633874E+06 2.633874E+06 -2.633874E+06 2.273447E+06 0.0 2.273447E+06 -2.273447E+06 3.018542E+05 3.018542E+05 -3.018542E+05

Static Aeroelastic Analysis Sample Problems

Figure 6-8. Output for Unit Trim Variable Solutions of Example HA144GB

NX Nastran Aeroelastic Analysis User’s Guide

6-143

Chapter

7 Flutter Analysis Sample Problems

•

Overview

•

Three Degree of Freedom Airfoil and Fuselage (Example HA145A)

•

Flutter Analysis of Jet Transport Wing by Lifting Surface Theory (Example HA145B)

•

Flutter and Divergence Analysis of Jet Transport Wing by Strip Theory (Example HA145C)

•

Flutter Analysis of Jet Transport Wing/Aileron by Strip Theory (Example HA145D)

•

Subsonic Flutter Analysis of the 15-Degree Sweptback Wing by the KE-Method (Example HA145E)

•

Low Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using Mach Box Aerodynamics (Example HA145F)

•

Low Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using ZONA51 Aerodynamics (Examples HA145FA and HA145FB)

•

High Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using Piston Theory (Example HA145G)

•

Flutter Analysis of a Square Simply Supported Panel (Examples HA145HA and HA145HB)

•

Estimation of Dynamic Stability Derivatives (Example HA145I)

•

Servoelastic Stability Analysis of a Missile (Example HA110A)

•

Aeroservoelastic Stability Analysis of a Missile (Example HA145J)

•

Aerothermoelastic Stability of a Wing (Examples HA153A and HA145KR)

NX Nastran Aeroelastic Analysis User’s Guide

7-1

Chapter 7

7.1

Flutter Analysis Sample Problems

Overview

The examples in this chapter illustrate a range of unsteady aerodynamic problems, primarily flutter analyses, using the three flutter methods and six aerodynamic methods available in NX Nastran. Estimation of aerodynamic static and dynamic stability derivatives and interactions with control systems and thermal loading are also demonstrated. Thirteen examples (or example pairs) are presented: •

Example HA145A presents a two-dimensional airfoil mounted on a fuselage with a plunging degree of freedom. This example illustrates the characteristics of the PK-method of flutter analysis for predicting both flutter and divergence of an unrestrained vehicle.

The next three examples are flutter analyses of the cantilevered BAH wing: •

Example HA145B and Example HA145C illustrate the PK- and K-methods for finding flutter and divergence speeds and also show significant differences in results between the Doublet-Lattice method (DLM) and Strip Theory.

•

Example HA145D includes the aileron using Strip Theory with an approximation to the circulation function and experimental correction factors for the spanwise loading.

The next four examples illustrate the effects of Mach number and demonstrate experimental correlation with wind-tunnel tests of a series of 15-deg swept-wing models: •

Example HA145E uses the KE-flutter method with DLM aerodynamics at a Mach number of m = 0.45.

•

Example HA145F uses the PK-method with supersonic Mach Box aerodynamics at m = 1.3.

•

Examples HA145FA and HA145FB reconsider Example HA145F using ZONA51 aerodynamics.

•

Example HA145G uses the PK-method with the high supersonic Piston Theory aerodynamics at m = 3.0.

The remaining examples present a variety of problems: •

Examples HA145HA and HA145HB perform supersonic panel flutter analyses of a square plate using Piston Theory and ZONA51 aerodynamics.

•

Example HA145I demonstrates the calculation of aerodynamic static and dynamic stability derivatives. Example HA145I applies the DLM to the forward-swept-wing (FSW) airplane to obtain aerodynamic coefficients for lateral-directional motions.

•

Example HA110A and Example HA145J illustrate the inclusion of a control system in two different stability analyses. These examples consider a simplified air-to-air missile, first without and then with the effects of an airstream, in representative servoelastic and aeroservoelastic stability analyses.

•

Examples HA153A and HA145KR present an aerothermoelastic analysis of a rectangular wing constructed and heated in a manner to emphasize thermoelastic effects on the aeroelastic characteristics. The solution requires sequential analyses of the nonlinear effects of temperature on the stiffness matrix and restarts in the aeroelastic analyses. Only the flutter analysis is demonstrated. However, the requirements for restarts for static and dynamic aeroelastic analyses are also discussed.

7-2

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

7.2 Three Degree of Freedom Airfoil and Fuselage (Example HA145A) A two degree of freedom (2 DOF) airfoil has been studied by Rodden and Bellinger (1982a) to illustrate the characteristics of the PK-flutter method. Later, a “fuselage” DOF in plunge was added to the problem by Rodden and Bellinger (1982b) to investigate the effects of a rigid body motion. The 3 DOF case is considered here. The airfoil is connected to the fuselage mass by bending and torsion springs, and the fuselage is free to plunge, as shown in Figure 7-1. The airfoil center of gravity may be at either 37% or 45% chord; its aerodynamic center is at 25% chord, and the springs are connected to the “elastic axis” at 40% chord. The remaining parameters for the analysis at sea level include a chord of 6.0 ft, a radius of gyration of 1.5 ft about the elastic axis, a mass ratio μ = 20.0, uncoupled bending and torsion frequencies of 10.0 and 25.0 rad/sec, respectively, and equal structural damping coefficients g = 0.03 in both modes.

Figure 7-1. Three Degree of Freedom Airfoil and Fuselage The airfoil lift curve slope is the theoretical two-dimensional incompressible value of C1 = 2π/rad. Both the exact Theodorsen function and an approximation are used for the circulation function. The approximation constants are those of W. P. Jones (1945): b0 = 1.0, b1 = 0.165, b2 = 0.335, β1 = 0.041, and β2 = 0.320. The mass ratio and the foregoing parameters translate to the following dimensional input parameters: Airfoil Mass = Fuselage Mass

=

Static Unbalance about Elastic Axis

={

Moment of Intertia about Elastic Axis

=

3.0256 slug-ft2

Bending Stiffness

=

134.47 lb/ft

Torsion Stiffness

=

1891.0 ft-lb/rad

Bending Damping

=

0.40341 lb-sec/ft

1.3447 slugs +0.24205 slug-ft for 37% c.g. -0.40342 slug-ft for 45% c.g.

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Torsion Damping

=

2.2692 ft-lb-sec/rad

The input data are set up with options for both centers of gravity and both aerodynamic representations. The option selected is the 37% chord center of gravity with the approximate Theodorsen function with the remaining options commented out using a “$” in the first column of the data entry.

Bulk Data Entries The Bulk Data entries specify that the two-dimensional airfoil has unit width. GRID 100 is on the airfoil centerline at the elastic axis with freedoms in plunge (T3) and pitch (R2). GRID 101 defines the elastic axis at the outboard edge of the strip and is connected to GRID 100 by RBAR 101. GRID 102 is the fuselage with only the plunge degree of freedom. CELAS2 103 is the bending spring connecting the airfoil elastic axis to the fuselage, and CELAS2 105 is the torsion spring between the airfoil elastic axis and ground (since the fuselage does not pitch). CMASS2 111 is the fuselage concentrated mass, and CONM1 110 gives the mass coupling of the airfoil. CDAMP2 123 is the bending damper between the airfoil and the fuselage, and CDAMP2 125 is the torsion damper to ground. ElGR selects the Modified Givens method MGlV for analysis of the vibration eigenvalues and eigenvectors. This completes the structural data. Next, the aerodynamic data are considered. The AERO entry includes a velocity of 100.0 fps (this is not needed for a flutter analysis but is required for a response analysis), the reference chord, sea-level density, and no symmetry or ground effect. The CAERO4 entry specifies Strip Theory for one strip with its leading edge 2.4 ft upstream from the elastic axis, a 6.0 ft chord and a unit width. Two PAERO4 entries are shown: one for the exact Theodorsen function and the other for the optional approximate circulation function. PAERO4 leads to the exact function by taking CLA = LCLA = CIRC = LClRC = 0, the absence of a control surface is indicated by DOC1 = CAOC1 = GAPO1 = 0.0, and no AEFACT entry is required. The optional PAERO4 specifies the lift curve slope of 2π via CLA = LCLA = 0, a two-term approximation via ClRC = 2 with the W. P. Jones coefficients on AEFACT 13 for Mach number, m = 0.0. This approximation to the circulation function is (Fung, 1969, p. 215):

Equation 7-1. A series of reduced frequencies for m = 0.0 is specified on two MKAERO1 entries from k = 0.001 to k = 5.0 in order to cover the frequency range of interest in the PK-flutter solution. The linear spline SPLlNE2 provides the data for the connection between the aerodynamics and the structure, which consists simply of aerodynamic element CAERO4 1000 and SET1 of grid points 100 and 101. The spline is rigidly attached to the elastic axis by selecting DTHX = DTHY = 0.0. FLFACT 1 gives the air density ratio and FLFACT 2 gives the Mach number. FLFACT 3 gives the series of velocities for flutter analysis by the PK-method; the negative values call for flutter modes (eigenvectors) as well as eigenvalues to be printed. The FLUTTER 3 entry specifies the British Flutter Method (PK). The FLUTTER 4 entry specifies the American Flutter Method (K), and FLFACT 4 provides the series of reduced frequencies required by the K-method. The ENDDATA entry ends the Bulk Data Section.

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Flutter Analysis Sample Problems

Case Control Commands In the Case Control Section, the title and label are shown along with the other options. ECHO = BOTH requests that the annotated Bulk Data Section with all comment entries be printed as well as the sorted data entries. METHOD = 1 specifies the eigenvalue method (ElGR entry) for the vibration analyses. SVEC = ALL requests the vibration modes printed for the degrees of freedom in the analysis set. DlSP = ALL requests the displacements at all grid points. The multiple subcase capability of SOL 145 is illustrated by requesting analyses by two flutter methods. In Subcase 1, FMETHOD = 3 specifies the PK-flutter method (FLUTTER entry); in Subcase 2, FLUTTER = 4 specifies the K-method, and the complex eigenvalue method by CMETHOD = 20 as required in the K-method. The remaining data specify the plotting formats for the V-g and V-f curves. BEGIN BULK is placed at the end of the Case Control Section. In the Executive Control Section, ID NXN, HA145A identifies this problem. TIME 5 restricts the run time to 5.0 CPU minutes. SOL 145 calls for the Aerodynamic Flutter DMAP Sequence, and CEND completes the Executive Control Section.

Output Both annotated and sorted input data are shown below in Listing 7-1 and Listing 7-2, respectively, and selected output follows in Listing 7-3. The plotted V-g and V-f curves for the PK-method are shown in Figure 7-2. The printed output begins with the vibration analysis which gives all three natural frequencies, generalized masses and vibration modes of the undamped, unrestrained system. The rigid body frequency is a computed zero (using the SUPORT entry would result in an exact zero). The output from the first subcase is the PK-flutter analysis. The first item is the zero frequency roots and is followed by the modal participation factors that were requested by SVEC and the negative velocities on the FLFACT entry. Next, the flutter summaries are presented, in which Point = No. refers to mode number for all velocities, followed by the flutter modes at the grid points requested by DISP and the negative velocities.

PK-Method Results The unique characteristics of the PK-method should be noted. In the case of the constrained fuselage (which is obtained by adding the T3 constraint to GRID 102 in this example), Rodden and Bellinger (1982a) showed that the PK-results were not continuous. A discontinuity occurs at some velocity (slightly different in the two representation of the Theodorsen function) at which the PK algorithm converges to an aerodynamic lag root (with zero frequency) rather than the bending root. The PK algorithm in NX Nastran then skips over the bending root in the reduced frequency “lining-up” process and next obtains the torsion root. In the unrestrained case (the present example), Rodden and Bellinger (1982b) did not find an obvious aerodynamic lag root because the PK solution appeared to be continuous, as is shown in Figure 7-2. The reason for the different behavior between restrained and unrestrained systems is a subject for further study. From Figure 7-2 and the printed Flutter Summary for Point 2, the “dynamic divergence” speed (i.e., the speed at which the oscillatory instability finds its origin in a tendency to static divergence) and frequency are 232.0 ft/s and 1.177 Hz, respectively. From Figure 7-2 and the Flutter Summary for Point 3, the flutter speed and frequency are 283.1 ft/s and 2.692 Hz, respectively. These results are only slightly different from those obtained by the “transient solution” in Rodden and Bellinger (1982b). Some explanation of the damping in Point 1 of the PK-Flutter Summary is also in order. The unrestrained system, free to plunge, has a zero eigenvalue as can be shown exactly with frequency-independent aerodynamics (i.e., with aerodynamic lags neglected). A frequency of zero

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Flutter Analysis Sample Problems

is given for Point 1 in the Summary, but the damping is not zero. The damping should be regarded as a computed zero and should not be interpreted as an instability between 120 and 140 ft/s.

Eigenvector Output The eigenvectors at speeds near divergence and flutter are also printed for the PK-method. The modal eigenvectors (modal participation factors) are shown before the Flutter Summaries and the physical eigenvectors are shown after the Summaries. In the physical eigenvectors, Point IDs 100 and 101 show the deflection and rotation of the airfoil elastic axis and Point ID 102 shows the fuselage deflection. The physical eigenvectors are not normalized.

K-Method Results Listing 7-3 concludes with Flutter Summaries for Subcase 2, the K-method; no modal data have been requested. The dynamic divergence speed and frequency are found by interpolating the second mode results to g = 0 between Points 6 and 7. The speed and frequency by linear interpolation are 231.9 ft/s and 1.196 Hz, respectively. The flutter speed and frequency are found by interpolating the third mode results between Points 2 and 3. The speed and frequency are 301.2 ft/s and 2.658 Hz, respectively. The interpolated results for the K-method differ slightly from the PK-method because of the wide spacing of the reduced frequencies in the K-method. A more refined K-method will lead to agreement with the PK-method.

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Figure 7-2. V-g and V-f Curves for Three Degree of Freedom Aeroelastic Model

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Flutter Analysis Sample Problems

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA145A $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145A $$$$$$$$ $ $ $ MODEL DESCRIPTION A TWO-DIMENSIONAL AIRFOIL IDEALIZED $ $ AS A SINGLE STRIP WITH A UNIT WIDTH, $ $ A MASS AT MID-SPAN WITH FREEDOM TO $ $ MOVE VERTICALLY RELATIVE TO A DUMB$ $ BELL AT MID-SPAN WITH A DISPLACEMENT $ $ SPRING BETWEEN THEM. THE DUMBBELL $ $ IS ATTACHED BY A TORSION SPRING TO $ $ GROUND. THIS RESULTS IN ONE RIGID $ $ BODY AND TWO FLEXIBLE DEGREES OF $ $ FREEDOM. THIS EXAMPLE HAS BEEN $ $ PUBLISHED IN THE JOURNAL OF AIRCRAFT, $ $ SEPT. 1982, PP. 796-797. $ $ $ $ SOLUTION PK FLUTTER ANALYSIS METHOD USING $ $ STRIP THEORY AERODYNAMICS WITH THE $ $ W.P.JONES APPROXIMATION TO THE $ $ THEODORSEN CIRCULATION FUNCTION. $ $ $ $ OUTPUT DISPLACEMENTS, AERODYNAMIC FORCES $ $ AND X-Y PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$ $$$$$$$$ TIME 5 $ TIME IN CPU MINUTES SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145A: THREE DEGREE OF FREEDOM AEROELASTIC MODEL PAGE 2 2-D AERO, W.P. JONES C(K) AND BRITISH (PK) FLUTTER METHOD ONE RIGID BODY AND TWO FLEXIBLE MODES, C.G. AT 0.37 CHORD C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145A: THREE DEGREE OF FREEDOM AEROELASTIC MODEL 2 SUBTI = 2-D AERO, W.P. JONES C(K) AND BRITISH (PK) FLUTTER METHOD 3 $SUBTI = 2-D AERO, EXACT C(K) AND BRITISH (PK) FLUTTER METHOD 4 LABEL = ONE RIGID BODY AND TWO FLEXIBLE MODES, C.G. AT 0.37 CHORD 5 $LABEL = ONE RIGID BODY AND TWO FLEXIBLE MODES, C.G. AT 0.45 CHORD 6 ECHO = BOTH 7 METHOD = 1 $ MODIFIED GIVENS EIGENVALUE METHOD 8 SVEC = ALL $ PRINT THE VIBRATION MODES 9 DISP = ALL $ PRINT THE FLUTTER MODES 10 SUBCASE 1 11 FMETHOD = 3 $ PK FLUTTER METHOD 12 SUBCASE 2 13 FMETHOD = 4 $ K FLUTTER METHOD 14 CMETHOD = 20 15 OUTPUT(XYOUT) 16 CSCALE 2.0 17 PLOTTER NASTRAN 18 CURVELINESYMBOL = -6 19 YTTITLE = DAMPING G 20 YBTITLE = FREQUENCY F HZ 21 XTITLE = VELOCITY V (FT/SEC) 22 XTGRID LINES = YES 23 XBGRID LINES = YES 24 YTGRID LINES = YES 25 YBGRID LINES = YES 26 UPPER TICS = -1 27 TRIGHT TICS = -1 28 BRIGHT TICS = -1 29 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 30 BEGIN BULK EXAMPLE HA145A: THREE DEGREE OF FREEDOM AEROELASTIC MODEL PAGE 3 2-D AERO, W.P. JONES C(K) AND BRITISH (PK) FLUTTER METHOD ONE RIGID BODY AND TWO FLEXIBLE MODES, C.G. AT 0.37 CHORD I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ * * * STRUCTURAL DATA * * * $ $ $ $ (LB-FT-SEC SYSTEM) $ $ * * GRID GEOMETRY * * $ $ $ $ GRID 102 (T3) SIMULATED FUSELAGE POINT $ $ GRID 100-101 (T3) WING POINTS $ $ $ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $

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Flutter Analysis Sample Problems

$ $ $ $ GRID GRID GRID $ $ $ $ $ $ $ $ $ $ $ $ CELAS2 CELAS2 $ $ $ $ $ $ $ $ RBAR $ $ $ $ $ $ $ $ $ CMASS2 $ $ $ $ $ $

ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS AND ITS ASSOCIATED SUPERELEMENT ID. ID 100 101 102

CP

X1 0. 0. 0.

X2 0.5 1.0 0.5

X3 0. 0. 0.

CD

PS 1246

$ $ $ SEID

12456

* * STRUCTURAL STIFFNESS PROPERTIES * * * SPRING MOUNTING DATA * THE CELAS2 ENTRY DEFINES A SCALAR SPRING ELEMENT WITHOUT REFERENCE TO A PROPERTY ENTRY, IN THIS CASE K IS THE SPRING CONSTANT, G1 AND G2 ARE JOINED BY THE SPRING (DEFAULT IS TO GROUND), C1 AND C2 ARE THE DOFS AT THESE POINTS, GE IS THE DAMPING COEFFICIENT AND S IS THE STRESS COEFFICIENT. POINT 102 MOVES RELATIVE TO POINT 100 LIKE A PISTON MOVES IN A CYLINDER. EID 103 105

K 134.47 1891.

G1 100 100

C1 3 5

G2 102

C2 3

GE

S

THE RBAR ENTRY DEFINES A RIGID BAR. LISTED ARE THE GRID POINTS AT EACH END AND THE DEPENDENT AND INDEPENDENT DOFS AT EACH END. THE NUMBER OF INDEPENDENT DOFS AT THE TWO ENDS MUST EQUAL SIX. BY DEFAULT THOSE NOT DECLARED INDEPENDENT ARE MADE DEPENDENT. EID 101

GA 100

GB 101

CNA 123456

CNB

CMA

M 1.3447

G1 102

C1 3

G2

$ $ $ $ $ $ $

CMB

* * MASS AND INERTIA PROPERTIES * * THE CMASS2 ENTRY DEFINES A SCALAR MASS ELEMENT WITHOUT REFERENCE TO A PROPERTY ENTRY. IT LISTS THE MASS, THE GRID NO. AND ITS DOF COMPONENTS. WHEN TWO GRID POINTS ARE LISTED THE MASS IS ADDED TO BOTH POINTS. * FUSELAGE MASS * EID 111

$ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $

C2

THE CONM1 ENTRY DEFINES A 6 BY 6 SYMMETRIC INERTIA MATRIX FOR A GRID POINT. LISTED IS THE ID, THE GRID POINT NO., THE COORDINATE SYSTEM IN WHICH THE INERTIA MATRIX IS DEFINED AND THE LOWER LEFT TRIANGULAR PART OF THE MATRIX. * WING MASS * $

$ $ EID G CID M11 M21 M22 M31 M32 CONM1 110 100 $ ELASTIC AXIS AT 37 PERCENT CHORD $ M33 M41 M42 M43 M44 M51 M52 M53 +CNM1 1.3447 0.24205 $ M54 M55 M61 M62 M63 M64 M65 M66 +CNM1A 3.0256 $ * * * $ ELASTIC AXIS AT 45 PERCENT CHORD $ M33 M41 M42 M43 M44 M51 M52 M53 $+CNM1 1.3447 -.40342 $ * * * $ $ $ THE CDAMP2 ENTRY DEFINES A SCALAR DAMPER ELEMENT WITHOUT REF$ ERENCE TO A PROPERTY ENTRY. B IS THE DAMPER CONSTANT, G1 AND $ G2 ARE THE GRID POINTS TO WHICH THE DAMPER IS ATTACHED AND $ C1 AND C2 ARE THE DOF COMPONENTS IN WHICH THEY MOVE. $ $ EID B G1 C1 G2 C2 CDAMP2 123 .40341 100 3 102 3 CDAMP2 125 2.2692 100 5 $ $ $ * * * AERODYNAMIC DATA * * * $ $ (LB-FT-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ REFERENCE LENGTHS PLUS SYMMETRY KEYS. SYMXZ = 0 INDICATES $ THAT THE MODEL IS MOUNTED WITH NO ROOT REFLECTION PLANE; $ SYMXY = 0 INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH $ FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 100. 6. .002378 0 0 $ $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * BEAM SPLINE FIT ON THE WING * $ $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT-

$ $ $ $ $ $ +CNM1 +CNM1A $ +CNM1A $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

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Flutter Analysis Sample Problems

Chapter 7

$

7-10

EID

$ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR $ ATTACHMENT AND TORSIONAL FLEXIBILITIES. DTHX AND DTHY $ ARE ROTATIONAL ATTACHMENT FLEXIBILITIES. CID IDENTIFIES $ THE SPLINE COORDINATE SYSTEM AND ITS AXIS. $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 1201 1000 1000 1000 1100 0. 1. 0 $ DTHX DTHY +SP2 0. 0. $ $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 ETC SET1 1100 100 101 $ $ THE CAERO4 ENTRY IS USED TO SPECIFY STRIP THEORY AERO. $ PID IDENTIFIES A PAERO4 ENTRY WHERE THE ELEMENT PROPERTIES $ ARE GIVEN, CP DEFAULTS TO THE BASIC COORDINATE SYSTEM, $ NSPAN IS THE NUMBER OF EQUAL SPAN STRIPS (IF UNEQUAL, $ LSPAN WOULD SPECIFY AN AEFACT ENTRY THAT WOULD DEFINE THE $ STRIPS WIDTHS). THE CONTINUATION ENTRY DEFINES THE INBD $ AND OUTBD LEADING EDGE POINTS AND THE CHORDS. $ PID CP NSPAN LSPAN CAERO4 1000 1000 1 $ X1 Y1 Z1 X12 X4 Y4 Z4 X43 +CA1 -2.4 0. 0. 6. -2.4 1. 0. 6. $ $ THE PAERO4 ENTRY DEFINES PROPERTIES OF THE STRIP ELEMENTS $ FOR STRIP THEORY AERODYNAMICS. IT OFFERS A WIDE VARIETY $ OF OPTIONS. CLA DEFAULTS TO NO COMPRESSIBILITY CORRECTION $ AND NO CORRECTIONS OF LIFT CURVE SLOPES. LCLA MUST BE ZERO $ WHEN CLA IS ZERO. CIRC=0 SPECIFIES THE EXACT EXPRESSION FOR $ THE THEODORSEN CIRCULATION FUNCTION. DOC, CAOC AND GAPOC $ ARE REQUIRED EVEN THOUGH THE STRIP HAS NO CONTROL SURFACE. $ IN FRACTIONS OF CHORD THESE ARE THE DISTANCE FROM THE QUARTER $ CHORD TO THE HINGE LINE, THE CONTROL SURFACE CHORD AND THE $ GAP BETWEEN THE PRIMARY SURFACE AND THE CONTROL SURFACE. $ CAOC1=0. SPECIFIES NO CONTROL SURFACE. $ $ PID CLA LCLA CIRC LCIRC DOC1 CAOC1 GAPOC1 PAERO4 1000 0 0 2 13 0. 0. 0. $ $ THIS AEFACT ENTRY AND ITS CONTINUATION CONTAIN SIX FIELDS $ FOR EACH MACH NO. ON THE MKAERO ENTRY; THE MACH NO. AND FIVE $ CONSTANTS USED TO APPROXIMATE THEODORSENS CIRCULATION FUNC$ TION. IN THIS CASE THE APPROXIMATION OF W.P. JONES IS USED $ AT MACH NO. 0.0. $ SID D1 D2 D3 ETC AEFACT 13 0.0 1.0 -0.165 0.041 -0.335 0.320 $ $ THE PAERO4 ENTRY DEFINES PROPERTIES OF THE STRIP ELEMENTS $ FOR STRIP THEORY AERODYNAMICS. IT OFFERS A WIDE VARIETY $ OF OPTIONS. CLA DEFAULTS TO NO COMPRESSIBILITY CORRECTION $ AND NO CORRECTIONS OF LIFT CURVE SLOPES. LCLA MUST BE ZERO $ WHEN CLA IS ZERO. CIRC=0 SPECIFIES THE EXACT EXPRESSION FOR $ THE THEODORSEN CIRCULATION FUNCTION. DOC, CAOC AND GAPOC $ ARE REQUIRED EVEN THOUGH THE STRIP HAS NO CONTROL SURFACE. $ IN FRACTIONS OF CHORD THESE ARE THE DISTANCE FROM THE QUARTER $ CHORD TO THE HINGE LINE, THE CONTROL SURFACE CHORD AND THE $ GAP BETWEEN THE PRIMARY SURFACE AND THE CONTROL SURFACE. $ CAOC1=0. SPECIFIES NO CONTROL SURFACE. $ $ AN ALTERNATE METHOD WOULD BE TO SET CIRC=0 TO SPECIFY THE $ EXACT EXPRESSION FOR THE THEODORSEN CIRCULATION FUNCTION $ (SEE EXAMPLE HA75BAHK). $ $ PID CLA LCLA CIRC LCIRC DOC1 CAOC1 GAPOC1 $PAERO4 1000 0 0 0 0 0. 0. 0. $ $ * * * SOLUTION SPECIFICATIONS * * * $ * * AERODYNAMIC CONDITIONS * * $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION WILL BE USED TO $ GENERATE GENERALIZED AERO FORCE MATRICES. IF MORE THAN EIGHT $ MACH NOS OR REDUCED FREQUENCIES ARE REQUIRED A SECOND MKAERO1 $ IS NECESSARY. MKAERO1 0. $ K1 K2 K3 K4 K5 ETC +MK .001 .1 .2 .3 .4 .5 .6 .7 MKAERO1 0. +MKA .8 .9 1. 1.1 1.3 1.5 2. 5. $ * * VIBRATION SOLUTION PARAMETERS * * $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM; IN THIS CASE THE $ MODIFIED GIVENS METHOD. THREE MODES ARE DESIRED, NORMAL-

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ +SP2 $ $ $ $ $ $ $ $ $ $ $ $ $ +CA1 $ $ $ $ $ $ $ $ $ $ $ $ $ +PAER $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +PAER $ $ $ $ $ $ $ $ $ +MK +MKA $ $ $ $

Flutter Analysis Sample Problems

$

$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 NO EIGR 1 MGIV 0. 25. 3 $ NORM G C +EGR MAX $ * * FLUTTER SOLUTION PARAMETERS * * $ $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE (DEFAULT IS 10-3). $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 3 PK 1 2 3 L 3 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ NEGATIVE VELOCITIES ARE SIGNALS TO COMPUTE AND PRINT EIGEN$ VECTORS. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 1. FLFACT 2 0. FLFACT 3 20. 40. 60. 80. 100. 120. 140. +FLF1 160. 170. 180. 190. 200. 210. 220. -230. +FLF2 -240. 260. -280. -290. 300. $ $ SID METHOD DENS MACH RFREQ IMETH NVALUE EPS FLUTTER 4 K 1 2 4 L 3 $ $ K1 K2 K3 K4 FLFACT 4 0.20 0.18 0.16 0.14 0.12 0.10 0.08 +FL4 0.06 $ $ THE EIGC ENTRY DEFINES DATA NEEDED TO PERFORM A COMPLEX $ EIGENVALUE ANALYSIS. THIS ONE SPECIFIES THAT THE UPPER $ HESSENBERG METHOD BE USED TO EXTRACT COMPLEX EIGENVALUES $ AND THAT THE EIGENVECTORS BE NORMALIZED ON THEIR MAXIMUM $ COMPONENTS. IF THEY WERE TO HAVE BEEN NORMALIZED ON A $ POINT THE GRID AND DOF COMPONENT WOULD HAVE BEEN REQUIRED. $ THE CONVERGENCE CRITERION DEFAULTS TO 1.0-4 FOR THE INVERSE $ POWER METHOD, TO 1.0-11 FOR THE DETERMINANT METHOD AND TO $ 1.0-15 FOR THE HESSENBERG METHOD. WHEN THE HESSENBERG $ METHOD IS SPECIFIED ALL ENTRIES ON THE CONTINUATION

$ $ $ +EGR $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO. +FLF1 +FLF2 VELOCITY $ $

+FL4 $ $ $ $ $ $ $ $ $ $

CARD $ $ EXCEPT THE NUMBER OF DESIRED ROOTS (NDL) ARE IGNORED. $ $ SID METHOD NORM G C E EIGC 20 HESS MAX $ ALPHAA1 OMEGAA1 ALPHAB1 OMEGAB1 L1 NEL NDL +EC 3 $ $ NOTE THE ABSENCE OF A PARAM,LMODES,N ENTRY TO STIPULATE THE $ NO. OF MODES TO BE USED IN THE ANALYSIS. LMODES DEFAULTS $ TO ZERO BUT THE PROGRAM THEN INTERROGATES LFREQ AND HFREQ $ WHICH DEFINE THE FREQUENCY RANGE OF INTEREST; LFREQ DEFAULTS $ TO ZERO AND HFREQ DEFAULTS TO 1.0E30. AS A RESULT ALL MODES $ ARE INCLUDED. $ ENDDATA INPUT BULK DATA CARD COUNT = 275

$ +EC

$ $ $ $ $ $ $

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Chapter 7

Listing 7-1. Input Files for Three Degree of Freedom Airfoil and Fuselage EXAMPLE HA145A: THREE DEGREE OF FREEDOM AEROELASTIC MODEL 2-D AERO, W.P. JONES C(K) AND BRITISH (PK) FLUTTER METHOD ONE RIGID BODY AND TWO FLEXIBLE MODES, C.G. AT 0.37 CHORD S O R T E D CARD COUNT 12345678910111213141516171819202122232425262728293031323334353637-

. 1 .. 2 .. 3 .. 4 AEFACT 13 0.0 1.0 AERO 100. 6. CAERO4 1000 1000 +CA1 -2.4 0. 0. CDAMP2 123 .40341 100 CDAMP2 125 2.2692 100 CELAS2 103 134.47 100 CELAS2 105 1891. 100 CMASS2 111 1.3447 102 CONM1 110 100 +CNM1 1.3447 +CNM1A 3.0256 EIGC 20 HESS MAX +EC EIGR 1 MGIV 0. +EGR MAX FLFACT 1 1. FLFACT 2 0. FLFACT 3 20. 40. +FLF1 160. 170. 180. +FLF2 -240. 260. -280. FLFACT 4 0.20 0.18 +FL4 0.06 FLUTTER 3 PK 1 FLUTTER 4 K 1 GRID 100 0. GRID 101 0. GRID 102 0. MKAERO1 0. +MKA .8 .9 1. MKAERO1 0. +MK .001 .1 .2 PAERO4 1000 0 0 RBAR 101 100 101 SET1 1100 100 101 SPLINE2 1201 1000 1000 +SP2 0. 0. ENDDATA TOTAL COUNT= 38

PAGE

B U L K

D A T A

.. 5 .. 6 -0.165 0.041 .002378 0 1 6. -2.4 3 102 5 3 102 5 3

9

E C H O

.. 7 .. 8 -0.335 0.320 0

..

9

..

10

.

+CA1 1. 3

0.

6.

3 +CNM1 0.24205 +CNM1A +EC 3

25.

3

+EGR DENSITY MACH NO. +FLF1 +FLF2 VELOCITY +FL4

60. 190. -290. 0.16

80. 200. 300. 0.14

100. 210.

120. 220.

140. -230.

0.12

0.10

0.08

2 2 0.5 1.0 0.5

3 4 0. 0. 0.

L L

3 3 1246

1.1

1.3

1.5

2.

5.

.3 2 123456

.4 13

.5 0.

.6 0.

.7 0.

1000

1100

0.

1.

0

12456 +MKA +MK

+SP2

Listing 7-2. Bulk Data Entries for Three Degree of Freedom Airfoil and Fuselage EXAMPLE HA145A: THREE DEGREE OF FREEDOM AEROELASTIC MODEL PAGE 2-D AERO, W.P. JONES C(K) AND BRITISH (PK) FLUTTER METHOD ONE RIGID BODY AND TWO FLEXIBLE MODES, C.G. AT 0.37 CHORD R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED NO. ORDER MASS 1 1 1.000000E-35 3.162278E-18 5.032922E-19 2.689400E+00 2 2 1.993302E+02 1.411843E+01 2.247019E+00 2.693529E+00 3 3 6.362626E+02 2.522425E+01 4.014564E+00 2.986609E+00 ONE RIGID BODY AND TWO FLEXIBLE MODES, C.G. AT 0.37 CHORD EIGENVALUE = 1.000000E-35 CYCLES = 5.032922E-19 R E A L E I G E N V E C T O R N O . 1 POINT ID. TYPE T1 100 G 102 G EIGENVALUE = 1.993302E+02 CYCLES = 2.247019E+00 POINT ID. TYPE 100 G 102 G EIGENVALUE = 6.362626E+02 CYCLES = 4.014564E+00

T2

R E A L

T1

E I G E N V E C T O R T2

R E A L

T3 1.000000E+00 1.000000E+00

R1

E I G E N V E C T O R

R1

N O .

GENERALIZED STIFFNESS 2.689400E-35 5.369017E+02 1.900268E+03 SUBCASE 1

R2 5.697271E-17

N O .

T3 -9.933019E-01 1.000000E+00

11

2 R2 -3.721132E-02

NX Nastran Aeroelastic Analysis User’s Guide

R3

3

POINT ID. TYPE T1 T2 T3 R1 R2 100 G -2.212632E-01 1.000000E+00 102 G 4.126024E-02 A ZERO FREQUENCY ROOT HAS EMERGED. WHEN THE MACH NO., DENSITY AND VELOCITY ARE COMPATIBLE IT MAY BE INTERPRETED TWO WAYS DEPENDING ON THE SIGN OF THE REAL PART: 1. (-) A MODE IS CRITICALLY DAMPED, OR,

7-12

R3

R3

Flutter Analysis Sample Problems

2. (+) THE SYSTEM IS DIVERGING. ONLY THE MOST CRITICAL ( I.E., MOST POSITIVE REAL ROOTS ) ARE PRINTED IN THE FLUTTER SUMMARY. FOR INFORMATIONAL PURPOSES, THE REMAINING REAL ROOTS ARE PRINTED HERE. LESS CRITICAL REAL ROOTS FOR LOOP 1 MACH VELOCITY DENSITY 0.00000E+00 2.00000E+01 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -1.4567375E-01 -3.3657432E-01 0.0000000E+00 0.0000 -8.9514378E-06 -2.0681997E-05 0.0000000E+00 LESS CRITICAL REAL ROOTS FOR LOOP 2 MACH VELOCITY DENSITY 0.00000E+00 4.00000E+01 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -1.5015207E-01 -6.9384265E-01 0.0000000E+00 0.0000 -8.9025161E-06 -4.1137933E-05 0.0000000E+00 LESS CRITICAL REAL ROOTS FOR LOOP 3 MACH VELOCITY DENSITY 0.00000E+00 6.00000E+01 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE * * * ETC * * * EIGENVECTOR FROM THE PK METHOD EIGENVALUE = -1.18322E-01 7.43723E+00 EIGENVECTOR 1.00000E+00 0.00000E+00 3.93887E-01 2.93080E-03 -5.80720E-02 1.29090E-02 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = 4.62476E-01 7.20415E+00 EIGENVECTOR 1.00000E+00 0.00000E+00 3.52272E-01 -7.41746E-02 -5.04782E-02 2.01211E-02 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = -4.15442E-02 1.69432E+01 EIGENVECTOR -2.99837E-01 2.79381E-02 1.00000E+00 0.00000E+00 1.06858E-01 -8.76371E-02 METHOD EIGENVALUE = 7.31108E-02 1.68646E+01 EIGENVECTOR -2.93904E-01 3.76281E-02 1.00000E+00 0.00000E+00 9.73343E-02 -8.69973E-02 FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0000 9.9999996E+24 2.0000000E+01 -8.9514378E-06 0.0000000E+00 -2.0681997E-05 0.0000000E+00 -4.1137933E-05 0.0000000E+00 0.0000 9.9999996E+24 4.0000000E+01 -8.9025161E-06 0.0000000E+00 0.0000 9.9999996E+24 6.0000000E+01 -8.8209827E-06 0.0000000E+00 -6.1141756E-05 0.0000000E+00 -8.0468417E-05 0.0000000E+00 0.0000 9.9999996E+24 8.0000000E+01 -8.7069448E-06 0.0000000E+00 0.0000 9.9999996E+24 1.0000000E+02 -8.5601641E-06 0.0000000E+00 -9.8889861E-05 0.0000000E+00 -1.1618088E-04 0.0000000E+00 0.0000 9.9999996E+24 1.2000000E+02 -8.3807663E-06 0.0000000E+00 0.0000 9.9999996E+24 1.4000000E+02 -8.1687558E-06 0.0000000E+00 -1.3211546E-04 0.0000000E+00 -1.4646759E-04 0.0000000E+00 0.0000 9.9999996E+24 1.6000000E+02 -7.9241345E-06 0.0000000E+00 0.0000 9.9999996E+24 1.7000000E+02 -7.7895957E-06 0.0000000E+00 -1.5297960E-04 0.0000000E+00 -1.5901127E-04 0.0000000E+00 0.0000 9.9999996E+24 1.8000000E+02 -7.6469050E-06 0.0000000E+00 0.0000 9.9999996E+24 1.9000000E+02 -7.4960758E-06 0.0000000E+00 -1.6453461E-04 0.0000000E+00 -1.6952117E-04 0.0000000E+00 0.0000 9.9999996E+24 2.0000000E+02 -7.3370970E-06 0.0000000E+00 0.0000 9.9999996E+24 2.1000000E+02 -7.1699874E-06 0.0000000E+00 -1.7394316E-04 0.0000000E+00 -1.7777071E-04 0.0000000E+00 0.0000 9.9999996E+24 2.2000000E+02 -6.9946791E-06 0.0000000E+00 0.0000 9.9999996E+24 2.3000000E+02 -6.8112399E-06 0.0000000E+00 -1.8097713E-04 0.0000000E+00 -1.8353370E-04 0.0000000E+00 0.0000 9.9999996E+24 2.4000000E+02 -6.6196476E-06 0.0000000E+00 0.0000 9.9999996E+24 2.6000000E+02 -6.2120389E-06 0.0000000E+00 -1.8658520E-04 0.0000000E+00 -1.8669791E-04 0.0000000E+00 0.0000 9.9999996E+24 2.8000000E+02 -5.7718062E-06 0.0000000E+00 0.0000 9.9999996E+24 2.9000000E+02 -5.5393807E-06 0.0000000E+00 -1.8557903E-04 0.0000000E+00 -1.8364380E-04 0.0000000E+00 0.0000 9.9999996E+24 3.0000000E+02 -5.2988953E-06 0.0000000E+00 POINT = 2 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 2.0932 4.7774005E-01 2.0000000E+01 -5.2922972E-02 2.2209418E+00 -3.6925915E-01 1.3954589E+01 -4.5262671E-01 1.3956034E+01 1.0467 9.5538121E-01 4.0000000E+01 -6.4864665E-02 2.2211716E+00 0.6975 1.4336574E+00 6.0000000E+01 -8.0392711E-02 2.2202647E+00 -5.6075263E-01 1.3950335E+01 -7.0381433E-01 1.3929142E+01 0.5223 1.9144514E+00 8.0000000E+01 -1.0105638E-01 2.2168918E+00 0.4165 2.4010513E+00 1.0000000E+02 -1.2927009E-01 2.2095175E+00 -8.9731598E-01 1.3882809E+01 -1.1579161E+00 1.3793537E+01 0.3448 2.8999088E+00 1.2000000E+02 -1.6789255E-01 2.1953096E+00 0.2923 3.4214358E+00 1.4000000E+02 -2.2941364E-01 2.1707933E+00 -1.5645435E+00 1.3639498E+01 -2.2812548E+00 1.3331023E+01 0.2500 4.0006928E+00 1.6000000E+02 -3.4224752E-01 2.1216981E+00 0.2288 4.3697596E+00 1.7000000E+02 -4.3594682E-01 2.0639074E+00 -2.8266604E+00 1.2967914E+01 -3.3707078E+00 1.2191103E+01 0.2032 4.9216213E+00 1.8000000E+02 -5.5297834E-01 1.9402742E+00 0.1739 5.7502460E+00 1.9000000E+02 -6.2116534E-01 1.7529358E+00 -3.4207640E+00 1.1014021E+01 -2.9245985E+00 9.6036949E+00 0.1441 6.9417725E+00 2.0000000E+02 -6.0905695E-01 1.5284754E+00 0.1187 8.4225025E+00 2.1000000E+02 -4.4093284E-01 1.3227477E+00 -1.8323117E+00 8.3110695E+00 -8.4656334E-01 7.7426677E+00 0.1056 9.4713259E+00 2.2000000E+02 -2.1867485E-01 1.2322838E+00 0.0970 1.0308497E+01 2.3000000E+02 -3.1818904E-02 1.1836718E+00 -1.1832226E-01 7.4372301E+00 4.6247628E-01 7.2041483E+00 0.0901 1.1104713E+01 2.4000000E+02 1.2839167E-01 1.1465758E+00 0.0784 1.2752671E+01 2.6000000E+02 4.0188041E-01 1.0816109E+00 1.3655820E+00 6.7959619E+00 2.0728457E+00 6.4386363E+00 0.0690 1.4495824E+01 2.8000000E+02 6.4387727E-01 1.0247407E+00 0.0648 1.5439999E+01 2.9000000E+02 7.6023901E-01 9.9643648E-01 2.3798504E+00 6.2607951E+00 2.6637485E+00 6.0808706E+00 0.0608 1.6445015E+01 3.0000000E+02 8.7610763E-01 9.6780056E-01 POINT = 3 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 3.6984 2.7039060E-01 2.0000000E+01 -6.2295899E-02 3.9240744E+00 -7.6797408E-01 2.4655687E+01 -1.1718910E+00 2.4523874E+01 1.8393 5.4368788E-01 4.0000000E+01 -9.5571436E-02 3.9030957E+00 1.2128 8.2455909E-01 6.0000000E+01 -1.3018502E-01 3.8603647E+00 -1.5788442E+00 2.4255388E+01

NX Nastran Aeroelastic Analysis User’s Guide

7-13

Flutter Analysis Sample Problems

Chapter 7

0.8947 1.1176720E+00 8.0000000E+01 -1.6754064E-01 3.7972960E+00 -1.9986857E+00 0.6992 1.4301561E+00 1.0000000E+02 -2.0801435E-01 3.7095006E+00 -2.4241452E+00 0.5641 1.7726365E+00 1.2000000E+02 -2.5113201E-01 3.5913720E+00 -2.8334291E+00 0.4623 2.1629729E+00 1.4000000E+02 -2.9648733E-01 3.4338062E+00 -3.1983929E+00 0.3808 2.6258354E+00 1.6000000E+02 -3.2693389E-01 3.2325952E+00 -3.3201764E+00 0.3459 2.8913383E+00 1.7000000E+02 -3.2582986E-01 3.1192405E+00 -3.1929317E+00 0.3141 3.1834149E+00 1.8000000E+02 -3.0907536E-01 2.9997017E+00 -2.9126773E+00 0.2872 3.4813454E+00 1.9000000E+02 -2.6960340E-01 2.8953786E+00 -2.4523396E+00 0.2668 3.7474556E+00 2.0000000E+02 -2.1918307E-01 2.8313422E+00 -1.9496168E+00 0.2510 3.9843569E+00 2.1000000E+02 -1.7416169E-01 2.7961466E+00 -1.5298979E+00 0.2376 4.2080059E+00 2.2000000E+02 -1.3744736E-01 2.7736089E+00 -1.1976544E+00 0.2259 4.4262891E+00 2.3000000E+02 -1.0691750E-01 2.7566836E+00 -9.2594594E-01 0.2155 4.6395979E+00 2.4000000E+02 -8.1843473E-02 2.7442884E+00 -7.0560837E-01 0.1972 5.0715075E+00 2.6000000E+02 -3.8695443E-02 2.7197886E+00 -3.3063197E-01 0.1815 5.5086040E+00 2.8000000E+02 -4.9039451E-03 2.6965926E+00 -4.1544240E-02 0.1745 5.7319441E+00 2.9000000E+02 8.6703515E-03 2.6840765E+00 7.3110797E-02 0.1678 5.9587369E+00 3.0000000E+02 1.9943634E-02 2.6709509E+00 1.6734783E-01 COMPLEX EIGENVALUE = -1.183223E-01, 7.437230E+00 C O M P L E X E I G E N V E C T O R NO. 3 (REAL/IMAGINARY) POINT ID. 100

TYPE G

T1 T2 T3 0.0 0.0 6.216006E-01 0.0 0.0 0.0 -5.767462E-03 0.0 101 G 0.0 0.0 6.216006E-01 0.0 0.0 0.0 -5.767462E-03 0.0 102 G 0.0 0.0 1.391491E+00 0.0 0.0 0.0 3.463432E-03 0.0 COMPLEX EIGENVALUE = 4.624763E-01, 7.204148E+00 C O M P L E X E I G E N V E C T O R (REAL/IMAGINARY)

R1

POINT ID. 100

T1 T2 T3 0.0 0.0 6.612567E-01 0.0 0.0 0.0 6.922573E-02 0.0 101 G 0.0 0.0 6.612567E-01 0.0 0.0 0.0 6.922573E-02 0.0 102 G 0.0 0.0 1.350189E+00 0.0 0.0 0.0 -7.334441E-02 0.0 COMPLEX EIGENVALUE = -4.154424E-02, 1.694319E+01 C O M P L E X E I G E N V E C T O R (REAL/IMAGINARY)

R1

POINT ID. 100

T1 T2 T3 0.0 0.0 -1.316783E+00 0.0 0.0 0.0 4.732900E-02 0.0 101 G 0.0 0.0 -1.316783E+00 0.0 0.0 0.0 4.732900E-02 0.0 102 G 0.0 0.0 7.045720E-01 0.0 0.0 0.0 2.432220E-02 0.0 COMPLEX EIGENVALUE = 7.311080E-02, 1.686455E+01 C O M P L E X E I G E N V E C T O R (REAL/IMAGINARY)

R1

POINT ID. 100

R1

TYPE G

TYPE G

TYPE G

101

G

102

G

T1 0.0 0.0 0.0 0.0 0.0 0.0

T2 0.0 0.0 0.0 0.0 0.0 0.0

T3 -1.308742E+00 5.687740E-02 -1.308742E+00 5.687740E-02 7.101119E-01 3.403857E-02

0.0 0.0 0.0 0.0 0.0 0.0

NO.

NO.

NO.

2.3859116E+01 2.3307480E+01 2.2565256E+01 2.1575241E+01 2.0310995E+01 1.9598766E+01 1.8847683E+01 1.8192202E+01 1.7789848E+01 1.7568708E+01 1.7427099E+01 1.7320755E+01 1.7242874E+01 1.7088936E+01 1.6943192E+01 1.6864552E+01 1.6782080E+01

R2 -7.272904E-02 1.279996E-02 -7.272904E-02 1.279996E-02 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0

7 R2 -6.358667E-02 2.288120E-02 -6.358667E-02 2.288120E-02 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0

12 R2 6.964681E-02 -8.763712E-02 6.964681E-02 -8.763712E-02 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0

16 R2 6.012296E-02 -8.699729E-02 6.012296E-02 -8.699729E-02 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0

FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.2000 5.0000000E+00 UNSTABL-SYSTEM UNSTABL-SYSTEM UNSTABL-SYSTEM 1.1734991E-11 -9.6750290E-12 0.2000 5.0000000E+00 1.8277669E+02 -2.8949064E-01 1.9393210E+00 -2.5155735E+01 1.7736099E+02 0.2000 5.0000000E+00 2.5796811E+02 -3.4388378E-02 2.7371268E+00 -4.4322772E+00 2.5785382E+02 *** USER INFORMATION MESSAGE 6142 (FA2) THE K OR KE METHOD OF FLUTTER ANALYSIS HAS FOUND A ROOT THAT HAS NO PHYSICAL INTERPRETATION. NOTE THAT TO OBTAIN PHYSICALLY MEANINGFUL V-G AND V-F PLOTS, XMIN,XMAX,YMIN,YMAX VALUES MUST BE EXPLICITLY SET IN THIS CASE. POINT = 2 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.1800 5.5555553E+00 0.0000000E+00 -1.7848259E-01 0.0000000E+00 -1.4121418E-15 1.5948895E-14 0.1800 5.5555553E+00 1.9263014E+02 -2.8513819E-01 1.8394824E+00 -2.6151154E+01 1.8708342E+02 0.1800 5.5555553E+00 2.8148340E+02 -1.1312733E-02 2.6879687E+00 -1.5920459E+00 2.8146988E+02 POINT = 3 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.1600 6.2500000E+00 0.0000000E+00 5.4728415E-02 0.0000000E+00 3.9312697E-16 -1.4377215E-14 0.1600 6.2500000E+00 2.0233513E+02 -2.6253778E-01 1.7174739E+00 -2.5474791E+01 1.9735406E+02 0.1600 6.2500000E+00 3.1127695E+02 5.7932930E-03 2.6422007E+00 9.0164042E-01 3.1127304E+02 POINT = 4 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.1400 7.1428571E+00 STABLE-SYSTEMSTABLE-SYSTEMSTABLE-SYSTEM-2.2613715E-11 -1.4847113E-11 0.1400 7.1428571E+00 2.1172806E+02 -2.1586071E-01 1.5725532E+00 -2.2209873E+01 2.0814951E+02 0.1400 7.1428571E+00 3.4997568E+02 1.6472060E-02 2.5993500E+00 2.8819215E+00 3.4994006E+02 *** USER INFORMATION MESSAGE 6142 (FA2) THE K OR KE METHOD OF FLUTTER ANALYSIS HAS FOUND A ROOT THAT HAS NO PHYSICAL INTERPRETATION. NOTE THAT TO OBTAIN PHYSICALLY MEANINGFUL V-G AND V-F PLOTS, XMIN,XMAX,YMIN,YMAX VALUES MUST BE EXPLICITLY SET IN THIS CASE. POINT = 5 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K

7-14

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

KFREQ 0.1200 0.1200 0.1200 POINT = KFREQ 0.1000 0.1000 0.1000 POINT = KFREQ 0.0800 0.0800 0.0800 POINT = KFREQ 0.0600 0.0600 0.0600

1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 8.3333340E+00 0.0000000E+00 4.1684758E-02 0.0000000E+00 4.5771654E-16 -2.1970395E-14 8.3333340E+00 2.2084439E+02 -1.3864943E-01 1.4059390E+00 -1.5128768E+01 2.1927432E+02 8.3333340E+00 4.0206161E+02 2.0703595E-02 2.5596037E+00 4.1609459E+00 4.0199698E+02 6 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.0000000E+01 0.0000000E+00 -1.2549019E-01 0.0000000E+00 -7.0328145E-16 -1.1252503E-14 1.0000000E+01 2.3030865E+02 -2.2779390E-02 1.2218254E+00 -2.6222949E+00 2.3026385E+02 1.0000000E+01 4.7557446E+02 1.9180996E-02 2.5230010E+00 4.5599475E+00 4.7550885E+02 7 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.2500000E+01 0.0000000E+00 1.2549019E-01 0.0000000E+00 5.8113581E-16 9.2981729E-15 1.2500000E+01 2.4225984E+02 1.4543825E-01 1.0281826E+00 1.7387835E+01 2.4036734E+02 1.2500000E+01 5.8628571E+02 1.3180855E-02 2.4882736E+00 3.8634534E+00 5.8624750E+02 8 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.6666668E+01 0.0000000E+00 -9.1076612E-01 0.0000000E+00 -6.9747285E-12 -1.8016331E-11 1.6666668E+01 2.6332538E+02 3.9871517E-01 8.3819067E-01 4.7855492E+01 2.4923715E+02 1.6666668E+01 7.7065411E+02 3.9374377E-03 2.4530683E+00 1.5171864E+00 7.7064960E+02

NX Nastran Aeroelastic Analysis User’s Guide

7-15

Chapter 7

Flutter Analysis Sample Problems

Listing 7-3. Output for Three Degree of Freedom Airfoil and Fuselage

7.3 Flutter Analysis of Jet Transport Wing by Lifting Surface Theory (Example HA145B) This example of the BAH wing is considered for its static aeroelastic characteristics in roll in Example HA144B and for its flutter characteristics using Strip Theory by Rodden (1959a) and Rodden, Harder, and Bellinger (1979). The present analysis illustrates the use of the subsonic Doublet-Lattice method (DLM) in a flutter analysis by the PK-method. The model is shown in Figure 7-3. For this example, the structural model is the same as discussed in Example HA144B except that the aileron and the CONM1 and CELAS2 Bulk Data entries that represented the model’s inertia and stiffness have been removed. The structural and mass data are contained in input files BAH_STRUCT.DAT (see Listing 6-4) and BAH_MASS.DAT (see Listing 6-5). The effect of structural damping, which was available in the dynamic problem, has been added via a PARAM,KDAMP,+1 entry and tabular values listed on a TABDMP1 entry. In this case, a zero value of structural damping is input for comparison with previous solutions to this problem that did not include structural damping a priori. The absence of the aileron also permits a simpler aerodynamic model than in Example HA144B. The aerodynamic model is divided into six strips across the span, as shown in Figure 7-3, and divided equally into four boxes chordwise. This aerodynamic idealization, while not representative of industrial practice, is consistent with the idealization of the structure into a small number of grid points. The new Bulk Data entries relate to the different aerodynamic idealization and the items required by the dynamic analysis, i.e., structural damping, and the vibration and flutter analyses. The PARAM KDAMP and TABDMP1 entries for structural damping have been discussed above. AERO specifies the aerodynamic coordinate system CORD2R 1 with its positive x-direction streamwise, the reference semichord, sea-level density, and symmetrical motion. The DLM aerodynamics are called for on CAERO1 1001, which calls for the body entry PAERO1 1000. (This entry is required even though no fuselage effects are considered.) The CAERO1 entry also includes: the basic coordinate system for the wing geometry, four equal chordwise divisions, AEFACT 77 to specify six unequal strips, and interference group IGlD 1. The continuation entry gives the geometrical data at the root and tip of the wing.

Splines The aerodynamic and structural grids are connected by the linear SPLlNE2 along the 35% chord elastic axis, which connects to all of the aerodynamic boxes outboard of the root strip, i.e., boxes 1005 through 1024. The spline is not connected to the root strip (boxes 1001 through 1004) since no motion of the fuselage is assumed, but the spline does go through all 11 structural grid points via SET1 14. The spline coordinate system CID is also the basic system because all geometry has been expressed relative to the elastic axis. The SPLlNE2 continuation entry sets DTHX = DTHY = -1.0 because there are no rotational degrees of freedom in the GENEL structural model. The aerodynamic and spline data are contained in the file BAH_AERO5.DAT and are shown in Listing 7-4. The vibration analysis uses EIGR 10, which requests the Modified Givens method to obtain all 10 modes normalized on the maximum deflection.

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Flutter Bulk Data Entries The modal flutter analysis requires a number of generalized aerodynamic influence coefficient matrices. These are obtained for the (m, k) pairs on the MKAERO1 entry. A Mach number of 0.0 and a reduced frequency range from 0.001 to 1.0 are chosen; selection of a very low reduced frequency in the PK-method is advised if divergence is to be determined for a restrained surface or if a low frequency root, e.g., the short period, is to be determined for an unrestrained vehicle. FLUTTER 40 selects the PK-method for a density ratio of 1.0 on FLFACT 1, a Mach number of 0.0 on FLFACT 2, and a series of velocities from 4800 in/s to 25,200 in/s on FLFACT 4 (flutter eigenvectors are requested on FLFACT 4 by negative velocities). Only linear interpolation on k of the (m,k) aerodynamic coefficients is allowed for the PK-method so the IMETH field is left blank, and five flutter roots are requested. The optional PARAM,LMODES entry uses all 10 vibration modes in the modal formulation so there is no problem with modal convergence when comparisons are made with earlier direct solutions. Finally, PARAM,VREF divides the solution velocities by 12.0 in/ft to output the velocities in units of ft/s. The ENDDATA entry completes the Bulk Data Section.

Case Control Commands The Case Control Section begins with the Title, Subtitle, and Label commands. ECHO = BOTH prints both annotated and sorted Bulk Data entries. SPC = 1 points to the Bulk Data entry SPC1 1, which adds the constraints in the plunge and pitch degrees of freedom to the fuselage. METHOD = 10 points to ElGR 10 for vibration analysis and SVEC = ALL requests the print of the vibration modes of the a-set grid points. FMETHOD = 40 points to the FLUTTER 40 entry for the PK-flutter method. DlSP = ALL prints all the displacements of the grid points in addition to the modal eigenvectors. The BEGIN BULK command ends the Case Control Section. The Executive Control Section gives the identification ID NXN, HA145B. TIME 10 specifies 10.0 minutes CPU computing time. SOL 145 calls for the Aerodynamic Flutter DMAP solution sequence. The remaining statements determine the formats of the plotted output. The CEND statement is last in the Executive Control Section.

Output The input data are shown in Listing 7-5 followed by the sorted Bulk Data entries in Listing 7-6. Selected items of the printed output are shown in Listing 7-7. The V-g and V-f curves are shown in Figure 7-4 and Figure 7-5. The significant results are discussed below. The first output shown is the weight and balance analysis. Next, the vibration analysis gives the 10 coupled frequencies of the cantilever wing. The first two frequencies are 12.798 rad/s = 2.037 Hz for first bending and 22.322 rad/s = 3.553 Hz for first torsion. These are in close agreement with the uncoupled bending and torsion frequencies of 12.799 and 22.357 rad/s obtained in Bisplinghoff, Ashley, and Halfman (1955, pp. 177 and 181). The first five vibration modes are given next. The PK-Flutter Summaries are preceded by the flutter modal participation factors and followed by the flutter grid point modes near the flutter and divergence speeds. The Flutter Summaries for the five requested flutter modes are presented, showing a number of instabilities, including both flutter and divergence. The V-f curves are shown in Figure 7-4 and the V-g curves are shown in Figure 7-5 for the first two aeroelastic modes. The lowest flutter speed is a coupled first-bending/ first-torsion mode flutter and is found in Figure 7-5 (and in Point 2 of the printed Flutter Summary by linear interpolation) to be 1056 ft/s = 720 mph. The divergence speed is found in Figure 7-5 and Point 1 of the Flutter Summary at 1651 ft/s = 1125 mph. The V-f curve for the bending mode in Figure 7-4 appears to be discontinuous as the frequency approaches zero near 1430 ft/s, and this behavior is similar to that observed by Rodden and Bellinger (1982a) using Strip Theory. However, a refined analysis in the range of velocities from 1420 to 1430 ft/s

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shows a continuous variation in the eigenvalue as it transitions from complex to real. There is no appearance of an aerodynamic lag root. Note in Figure 7-5 that the definition of damping changes between the low speed branch, with its finite frequency, and the high speed branch, with its zero frequency.

Figure 7-3. BAH Wing Planform and Aerodynamic Strip Idealization

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Figure 7-4. V-f Curve for BAH Wing

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Figure 7-5. V-g Curve for BAH Wing $ $ $ $ $ $ $ $ CORD2R $ +C1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ CAERO1 $ +CA1 $ $

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THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM FLAGGED BY THE AERO ENTRY. LISTED ARE THE ORIGIN, A POINT ALONG THE Z AXIS AND A POINT IN THE X-Z PLANE, ALL IN THE RID COORDINATE SYSTEM. NOTE THAT IN THIS COORDINATE SYSTEM THE UPSIDE DOWN AND BACKWARDS STRUCTURE WILL FLY UPSIDE UP AND FORWARD. CID 1 C1 -1.

RID C2 0.

A1 0. C3 0.

A2 0.

A3 0.

B1 0.

B2 0.

$ $ $ $ $ $ $ B3 -1.

THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AERODYNAMICS. LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE USED TO PARTITION THE WING INTO AERODYNAMIC BOXES, THE FORMER FOR UNIFORMLY SPACED BOXES AND THE LATTER FOR NON-UNIFORMLY SPACED BOXES. IGID IS THE ID OF ITS ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY DEFINES POINTS 1 AND 2, THE ROOT CHORD AND THE TIP CHORD. THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED BEGINNING WITH EID, SO A NUMBER SHOULD BE CHOSEN THAT IS UNIQUE, AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND EXTRA POINT IDS. EID PID CP 1001 1000 0 (FWD INBOARD POINT) 78.75 0.0 0.0

LCHORD IGID NCHORD LSPAN 4 77 1 ROOTCHORD (FWD OUTBOARD POINT) TIP CHORD 35.0 500.0 0.0 100.0 225.0 NSPAN

THE AEFACT ENTRY IS A UTILITY ENTRY USED TO SPECIFY LISTS OF

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+C1

$ $ $ $ $ $ $ $ $ $ $ $ $ $ +CA +CA1

$ $

Flutter Analysis Sample Problems

$ $ $ $ $ AEFACT $ $ $ $ PAERO1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ SPLINE2 $ +SP100 $ $ $ $ $ SET1

NUMBERS. THE CAERO1 ENTRY IDENTIFIES THEM BY LSPAN AND LCHORD. $ THIS AEFACT ENTRY AND ITS CONTINUATION ENTRY CONTAIN SIX FIELDS $ WHICH SPECIFY THE STRIP EDGES AS FRACTIONS OF THE SPAN. $ $ SID D1 D2 D3 ETC 77 .0 .09 .276 .454 .636 .826 1.0 $ THE PAERO ENTRY IS REQUIRED EVEN THOUGH IT IT NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ 1000 $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * LINEAR SPLINE FIT ON THE WING * $ $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT$ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR $ ATTACHMENT AND TORSIONAL FLEXIBILITIES. DTHX AND DTHY $ ARE ROTATIONAL ATTACHMENT FLEXIBILITIES. CID IDENTIFIES $ THE SPLINE AXIS. $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID 100 1001 1005 1024 14 0.0 1.0 0 +SP100 DTHX DTHY -1.0 -1.0 $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 G5 G6 14 1 THRU 11

Listing 7-4. Input File BAH_AERO5.DAT N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA145B $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145B $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ $ CANTILEVERED WING WITH TEN BEAM $ $ ELEMENTS AND DUMBBELL MASSES $ $ $ $ SOLUTION PK FLUTTER ANALYSIS METHOD USING $ $ DOUBLET-LATTICE AERODYNAMICS AT $ $ MACH NO. 0.0 $ $ $ $ $ $ OUTPUT TABULATED MODAL DEFLECTIONS PLUS $ $ X-Y PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ TIME 10 $ TIME IN CPU MINUTES SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145B: BAH JET TRANSPORT WING FLUTTER ANALYSIS PAGE 2 CANTILEVERED, DOUBLET-LATTICE AERODYNAMICS AT MACH NO. 0.0 PK FLUTTER METHOD C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145B: BAH JET TRANSPORT WING FLUTTER ANALYSIS 2 SUBTI = CANTILEVERED, DOUBLET-LATTICE AERODYNAMICS AT MACH NO. 0.0 3 LABEL = PK FLUTTER METHOD 4 ECHO = BOTH 5 SPC = 1 $ FUSELAGE CONSTRAINT 6 SDAMP = 2000 $ STRUCTURAL DAMPING 7 METHOD = 10 $ MODIFIED GIVENS FOR VIBRATION ANALYSIS 8 SVEC = ALL $ PRINT VIBRATION MODES

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9 FMETHOD = 40 $ PK-FLUTTER METHOD 10 DISP = ALL $ PRINT FLUTTER MODES 11 OUTPUT(XYOUT) 12 CSCALE 2.0 13 PLOTTER NASTRAN 14 CURVELINESYMBOL = -6 15 YTTITLE = DAMPING G 16 YBTITLE =FREQUENCY F HZ 17 XTITLE = VELOCITY V (FT/SEC) 18 XMIN = 0. 19 XMAX = 2500. 20 YTMIN = -.6 21 YTMAX = +.1 22 YBMIN = 0. 23 YBMAX = 15. 24 XTGRID LINES = YES 25 XBGRID LINES = YES 26 YTGRID LINES = YES 27 YBGRID LINES = YES 28 UPPER TICS = -1 29 TRIGHT TICS = -1 30 BRIGHT TICS = -1 31 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 4(G,F) 5(G,F) 32 CURVELINESYMBOL = -6 33 YTTITLE = DAMPING G 34 YBTITLE =FREQUENCY F HZ 35 XTITLE = VELOCITY V (FT/SEC) 36 XMIN = 0. 37 XMAX = 1600. 38 YTMIN = -1.7 39 YTMAX = +.1 40 YBMIN = 0. 41 YBMAX = 2.5 42 XTGRID LINES = YES 43 XBGRID LINES = YES 44 YTGRID LINES = YES 45 YBGRID LINES = YES 46 UPPER TICS = -1 47 TRIGHT TICS = -1 48 BRIGHT TICS = -1 49 XYPLOT VG / 1(G,F) 50 BEGIN BULK EXAMPLE HA145B: BAH JET TRANSPORT WING FLUTTER ANALYSIS CANTILEVERED, DOUBLET-LATTICE AERODYNAMICS AT MACH NO. 0.0 PK FLUTTER METHOD I N P U T B U L K D A T A D E C K E C H O

PAGE

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ INCLUDE BAH_STRUCT.DAT $ INCLUDE BAH_MASS.DAT $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $ $ THE SPC1 ENTRY CONSTRAINS DOFS OF THE LISTED GRID POINTS. $ $ $ $ SID C G1 G2 ETC. SPC1 1 345 11 $ $ $ * * STRUCTURAL DAMPING * * $ $ $ $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUSION $ $ OF STRUCTURAL DAMPING IN EQUATIONS OF MOTION (SEE HANDBOOK $ $ FOR DYNAMIC ANALYSIS, SECT. 3.2.2). IF SET TO -1, MODAL $ $ DAMPING IS PUT INTO COMPLEX STIFFNESS MATRIX AS STRUCTURAL $ $ DAMPING. $ $ $ $ N V1 V2 PARAM KDAMP +1 $ $ $ THE TABDMP1 ENTRY DEFINES MODAL DAMPING AS A TABULAR $ $ FUNCTION OF FREQUENCY. THE DAMPING LEVELS ARE LINEAR $ $ BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE ZERO $ $ OUTSIDE THE TABULATED FREQUENCY RANGE. $ $ $ $ ID +TDP TABDMP1 2000 +T2000 $ F1 G1 F2 G2 ETC ENDT +T2000 0.0 0.0 10.0 0.0 ENDT $ $ $ $ $ * * * AERODYNAMIC DATA * * * $ $ $

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$ (SNAIL-IN-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD $ AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=1 INDICATES $ THAT THE MODEL IS MOUNTED WITH A ROOT REFLECTION PLANE; $ SYMXY = 0 INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH $ FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 131.232 1.1468-7 1 $ INCLUDE BAH_AERO5.DAT $ $ * * * SOLUTION SPECIFICATIONS * * * $ $ * VIBRATION SOLUTION PARAMETERS * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM, IN THIS CASE THE $ MODIFIED GIVENS METHOD. TEN MODES ARE DESIRED, NORMALIZED $ ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 ND EIGR 10 MGIV 10 $ NORM G C +EIGR MAX $ $ * AERODYNAMIC CONDITIONS * $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ TO GENERATE GENERALIZED AERO FORCE MATRICES. IF MORE THAN $ EIGHT MACH NO.S OR REDUCED FREQUENCIES ARE REQUIRED A SECOND $ MKAERO1 ENTRY IS NECESSARY. $ $ M1 M2 M3 ETC MKAERO1 0. $ K1 K2 K3 K4 K5 ETC +MK 0.001 0.05 0.10 0.20 0.50 1.0 $ $ *FLUTTER SOLUTION PARAMETERS * $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE (DEFAULT IS 10-3). $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 40 PK 1 2 4 L 5 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NO.S $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ NEGATIVE VELOCITIES ARE SIGNALS TO COMPUTE AND PRINT EIGEN$ VECTORS. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 1. FLFACT 2 .0 FLFACT 4 4800. 6000. 7200. 8400. 9600. 10800. -12000. +FLF4 -13200. 14400. 15600. 16800. 16920. 17040. 17100. 17112. +FLF4A 17124. 17136. 17148. 17160. 18000. -19200. -20400. 21600. +FLF4B 22800. 24000. 25200. $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. $ PARAM LMODES 10 $ $ THE PARAM,VREF,C ENTRY SPECIFIES A CONVERSION FACTOR TO BE $ USED TO CONVERT THE DIMENSIONS OF THE OUTPUT VELOCITIES BY $ DIVIDING BY C, IN THIS CASE BY 12.0 IN/FT TO PRINT VEL$ OCITIES IN FT/SEC RATHER THAN IN/SEC. $ PARAM VREF 12.0 $ ENDDATA INPUT BULK DATA CARD COUNT = 333

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +EIGR $ $ $ $ $ $ $ $ $ $ +MK $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO +FLF4 +FLF4A +FLF4B VELOCITY $ $ $ $ $ $ $ $ $ $

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Listing 7-5. Input Files for Jet Transport Wing by Lifting Surface Theory EXAMPLE HA145B: BAH JET TRANSPORT WING FLUTTER ANALYSIS PAGE 11 CANTILEVERED, DOUBLET-LATTICE AERODYNAMICS AT MACH NO. 0.0 PK FLUTTER METHOD S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 77 .0 .09 .276 .454 .636 .826 1.0 2AERO 1 131.232 1.1468-71 3CAERO1 1001 1000 0 4 77 1 +CA1 4+CA1 78.75 0.0 0.0 225.0 35.0 500.0 0.0 100.0 5CMASS2 121 5248.7 1 3 6CMASS2 122 134.9 1 3 2 3 7CMASS2 123 790.3 2 3 8CMASS2 341 9727. 3 3 9CMASS2 342 11005. 3 3 4 3 10CMASS2 343 473. 4 3 11CMASS2 561 3253.6 5 3 12CMASS2 562 -139.7 5 3 6 3 13CMASS2 563 946.3 6 3 14CMASS2 781 2617.8 7 3 15CMASS2 782 21. 7 3 8 3 16CMASS2 783 782.3 8 3 17CMASS2 9101 494.8 9 3 18CMASS2 9102 -7.3 9 3 10 3 19CMASS2 9103 185.2 10 3 20CONM1 1 11 +51 21+51 17400. 4.37+7 +52 22+52 4.35+09 23CORD2R 1 0. 0. 0. 0. 0. -1. +C1 24+C1 -1. 0. 0. 25EIGR 10 MGIV 10 +EIGR 26+EIGR MAX 27FLFACT 1 1. DENSITY 28FLFACT 2 .0 MACH NO 29FLFACT 4 4800. 6000. 7200. 8400. 9600. 10800. -12000. +FLF4 30+FLF4 -13200. 14400. 15600. 16800. 16920. 17040. 17100. 17112. +FLF4A 31+FLF4A 17124. 17136. 17148. 17160. 18000. -19200. -20400. 21600. +FLF4B 32+FLF4B 22800. 24000. 25200. VELOCITY 33FLUTTER 40 PK 1 2 4 L 5 34GENEL 432 1 3 2 3 3 3 +01 35+01 4 3 5 3 6 3 7 3 +02 36+02 8 3 9 3 10 3 +03 37+03 UD 11 3 11 4 11 5 +04 38+04 11 6 +05 39+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 40+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 41+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 42+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 43+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 44+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 45+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 46+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 47+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 48+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 49+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 50+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 51+17 413.0 52GRID 1 20.25 90. 12456 53GRID 2 -81. 90. 12456 54GRID 3 17.85 186. 12456 55GRID 4 -71.4 186. 12456 56GRID 5 15.8 268. 12456 57GRID 6 -63.2 268. 12456 58GRID 7 13.3 368. 12456 59GRID 8 -53.2 368. 12456 60GRID 9 11.05 458. 12456 61GRID 10 -44.2 458. 12456 62GRID 11 0.0 0. 126 63MKAERO1 0. +MK 64+MK 0.001 0.05 0.10 0.20 0.50 1.0 65PAERO1 1000 66PARAM GRDPNT 11 67PARAM KDAMP -1 68PARAM LMODES 10 69PARAM VREF 12.0 70PARAM WTMASS .0025907 71SET1 14 1 THRU 11 72SPC1 1 345 11 73SPLINE2 100 1001 1005 1024 14 0.0 1.0 0 +SP100 74+SP100 -1.0 -1.0 75TABDMP1 2000 +T2000 76+T2000 0.0 0.0 10.0 0.0 ENDT ENDDATA

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TOTAL COUNT=

77

Listing 7-6. Sorted Bulk Data Entries for Jet Transport Wing by Lifting Surface Theory EXAMPLE HA145B: BAH JET TRANSPORT WING FLUTTER ANALYSIS PAGE 14 CANTILEVERED, DOUBLET-LATTICE AERODYNAMICS AT MACH NO. 0.0 PK FLUTTER METHOD O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 11 M O * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 4.191900E+04 5.128960E+06 -1.642074E+05 0.000000E+00 * * 0.000000E+00 0.000000E+00 5.128960E+06 1.350243E+09 -2.381847E+07 0.000000E+00 * * 0.000000E+00 0.000000E+00 -1.642074E+05 -2.381847E+07 4.458782E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Y 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Z 4.191900E+04 3.917256E+00 1.223541E+02 0.000000E+00 I(S) * 7.226942E+08 3.727022E+06 0.000000E+00 * * 3.727022E+06 4.458139E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 * I(Q) * 4.458143E+09 * * 7.226906E+08 * * 0.000000E+00 * Q * 9.977437E-04 9.999995E-01 0.000000E+00 * * -9.999995E-01 9.977437E-04 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 1 1.637768E+02 1.279753E+01 2.036790E+00 8.160930E+00 1.336571E+03 2 2 4.982469E+02 2.232145E+01 3.552568E+00 5.525822E+01 2.753224E+04 3 3 2.092550E+03 4.574440E+01 7.280447E+00 7.079897E+00 1.481504E+04 4 4 5.402874E+03 7.350424E+01 1.169856E+01 8.652712E+00 4.674951E+04 5 5 8.742090E+03 9.349915E+01 1.488085E+01 4.002357E+00 3.498896E+04 6 8 1.766007E+04 1.328912E+02 2.115029E+01 3.883447E+00 6.858195E+04 7 6 2.398459E+04 1.548696E+02 2.464826E+01 3.597392E+00 8.628196E+04 8 10 4.211864E+04 2.052283E+02 3.266309E+01 3.142601E+00 1.323621E+05 9 9 6.020812E+04 2.453734E+02 3.905239E+01 1.016253E+00 6.118669E+04 10 7 9.183204E+04 3.030380E+02 4.823000E+01 8.617019E+00 7.913184E+05 EIGENVALUE = 1.637768E+02 CYCLES = 2.036790E+00 R E A L E I G E N V E C T O R N O . 1 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 4.518474E-02 2 G 4.250221E-02 3 G 1.482559E-01 4 G 1.437271E-01 5 G 3.794104E-01 6 G 3.767312E-01 7 G 6.929946E-01 8 G 6.926546E-01 9 G 9.989689E-01 10 G 1.000000E+00 EIGENVALUE = 4.982469E+02 CYCLES = 3.552568E+00 R E A L E I G E N V E C T O R N O . 2 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G -1.635563E-01 2 G 5.449901E-01 3 G -2.796319E-01 4 G 1.000000E+00 5 G -2.267440E-01 6 G 9.465655E-01 7 G -1.388892E-01 8 G 8.878612E-01 9 G -3.973597E-02 10 G 8.224477E-01 EIGENVALUE = 2.092550E+03 CYCLES = 7.280447E+00 POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G

R E A L T2

E I G E N V E C T O R T3 -2.196879E-01 -2.438909E-01 -3.054668E-01 -3.628896E-01 -3.718194E-01 -4.285979E-01 2.038223E-01 1.633554E-01 1.000000E+00

N O . R1

3 R2

R3

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10 EIGENVALUE = CYCLES =

G 5.402874E+03 1.169856E+01

9.818934E-01 R E A L

POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G EIGENVALUE = 8.742090E+03 CYCLES = 1.488085E+01 POINT ID. 1 2 3 4 5 6 7 8 9 10

TYPE G G G G G G G G G G

EIGENVALUE = EIGENVECTOR

R E A L

T1

T3 2.913901E-01 2.796087E-01 3.067727E-01 3.535685E-01 -5.594192E-01 -4.900773E-01 -1.464054E-01 -2.030947E-02 8.377002E-01 1.000000E+00 E I G E N V E C T O R

4

R1

N O .

T3 R1 -1.906066E-04 -7.756479E-02 -4.789248E-03 -9.476011E-02 -9.394086E-02 4.210923E-01 -3.057272E-01 1.000000E+00 -3.756571E-01 9.481917E-01 EIGENVECTOR FROM THE PK METHOD 1.96569E+01

R2

R2

EIGENVECTOR FROM THE PK METHOD 1.92174E+01

0.00000E+00 -6.34056E-02 -2.27926E-02 -4.07515E-04 -3.33487E-03 -8.20808E-04 -1.45133E-03 7.91778E-04 -6.62259E-05 1.04548E-04

A ZERO FREQUENCY ROOT HAS EMERGED. WHEN THE MACH NO., DENSITY AND VELOCITY ARE COMPATIBLE IT MAY BE INTERPRETED TWO WAYS DEPENDING ON THE SIGN OF THE REAL PART: 1. (-) A MODE IS CRITICALLY DAMPED, OR, 2. (+) THE SYSTEM IS DIVERGING. ONLY THE MOST CRITICAL ( I.E., MOST POSITIVE REAL ROOTS ) ARE PRINTED IN THE FLUTTER SUMMARY. FOR INFORMATIONAL PURPOSES, THE REMAINING REAL ROOTS ARE PRINTED HERE. LESS CRITICAL REAL ROOTS FOR LOOP 19 MACH VELOCITY DENSITY 0.00000E+00 1.71600E+04 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -1.0980374E-01 -9.9521217E+00 0.0000000E+00 0.0000 -9.3707547E-02 -8.4932337E+00 0.0000000E+00 LESS CRITICAL REAL ROOTS FOR LOOP 20 MACH VELOCITY DENSITY 0.00000E+00 1.80000E+04 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -1.7721196E-01 -1.6847939E+01 0.0000000E+00 0.0000 -3.9179351E-02 -3.7248688E+00 0.0000000E+00 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = -1.00246E+00 0.00000E+00 EIGENVECTOR 1.00000E+00 0.00000E+00 -1.93432E-02 0.00000E+00 1.25923E-02 0.00000E+00 5.83946E-03 0.00000E+00 -1.55428E-02 0.00000E+00 4.48947E-03 0.00000E+00 -1.11579E-03 0.00000E+00 -8.13973E-04 0.00000E+00 -3.60419E-03 0.00000E+00 2.51953E-04 0.00000E+00 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = 1.02447E+00 0.00000E+00 EIGENVECTOR 1.00000E+00 0.00000E+00 -1.93953E-02 0.00000E+00 1.07119E-02 0.00000E+00 5.81710E-03 0.00000E+00

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R3

5

0.00000E+00 -6.91265E-02 -2.10305E-02 -7.97172E-04 -3.57499E-03 -9.72727E-04 -1.52440E-03 7.81682E-04 -1.17154E-04 7.67171E-05

1.01338E-01

1.00000E+00 8.18942E-02 -8.22495E-03 -7.60734E-03 1.82604E-02 -5.27874E-03 1.16707E-03 7.91223E-04 3.96970E-03 -3.58191E-04

N O .

T2

-1.31927E-01

1.00000E+00 1.13370E-01 -6.34859E-03 -8.14869E-03 1.99117E-02 -5.61990E-03 1.38024E-03 7.52492E-04 4.24761E-03 -3.95885E-04 EIGENVALUE = EIGENVECTOR

T2

E I G E N V E C T O R

R3

Flutter Analysis Sample Problems

-1.51907E-02 4.39807E-03 -1.15492E-03 -7.33074E-04 -3.48011E-03 2.62640E-04 POINT = KFREQ 0.1737 0.1387 0.1152 0.0981 0.0849 0.0739 0.0642 0.0549 0.0452 0.0338 0.0161 0.0131 0.0090 0.0060 0.0052 0.0043 0.0030 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 POINT = KFREQ 0.2995 0.2370 0.1947 0.1641 0.1407 0.1223 0.1075 0.0955 0.0858 0.0777 0.0708 0.0702 0.0696 0.0693 0.0692 0.0692 0.0691 0.0690 0.0690 0.0650 0.0598 0.0553 0.0514 0.0478 0.0446 0.0418 POINT KFREQ 0.6196 0.4958 0.4133 0.3542 0.3099 0.2754 0.2477 0.2250 0.2060 0.1898 0.1757 0.1744 0.1731 0.1725 0.1724 0.1722 0.1721 0.1720 0.1719 0.1633 0.1522 0.1420 0.1324 0.1231 0.1142 0.1059

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

FLUTTER SUMMARY MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 5.7573352E+00 4.0000000E+02 -1.7215297E-01 2.0222278E+00 -1.0936906E+00 1.2706033E+01 7.2089267E+00 5.0000000E+02 -2.2807270E-01 2.0187895E+00 -1.4464860E+00 1.2684429E+01 8.6812620E+00 6.0000000E+02 -2.9225904E-01 2.0116851E+00 -1.8470465E+00 1.2639791E+01 1.0193741E+01 7.0000000E+02 -3.6837316E-01 1.9987390E+00 -2.3130975E+00 1.2558448E+01 1.1784474E+01 8.0000000E+02 -4.6180311E-01 1.9759294E+00 -2.8666730E+00 1.2415131E+01 1.3524364E+01 9.0000000E+02 -5.7986718E-01 1.9369451E+00 -3.5285454E+00 1.2170185E+01 1.5566833E+01 1.0000000E+03 -7.3513758E-01 1.8697836E+00 -4.3182707E+00 1.1748198E+01 1.8206732E+01 1.1000000E+03 -9.4788873E-01 1.7585402E+00 -5.2367225E+00 1.1049234E+01 2.2124870E+01 1.2000000E+03 -1.2661015E+00 1.5786728E+00 -6.2792897E+00 9.9190941E+00 2.9572548E+01 1.3000000E+03 -1.8537142E+00 1.2795175E+00 -7.4514174E+00 8.0394459E+00 6.2085567E+01 1.4000000E+03 -4.2601905E+00 6.5634024E-01 -8.7843161E+00 4.1239076E+00 7.6550934E+01 1.4100000E+03 -5.3009262E+00 5.3611785E-01 -8.9281588E+00 3.3685279E+00 1.1090582E+02 1.4200000E+03 -7.7507062E+00 3.7267104E-01 -9.0743771E+00 2.3415613E+00 1.6583479E+02 1.4250000E+03 -1.1643057E+01 2.5010988E-01 -9.1484547E+00 1.5714867E+00 1.9121582E+02 1.4260000E+03 -1.3437467E+01 2.1706378E-01 -9.1633577E+00 1.3638520E+00 2.3360669E+02 1.4270000E+03 -1.6431654E+01 1.7779942E-01 -9.1782846E+00 1.1171467E+00 3.2805820E+02 1.4280000E+03 -2.3096727E+01 1.2669776E-01 -9.1932545E+00 7.9606557E-01 2.1647014E+03 1.4290000E+03 -1.5254637E+02 1.9214356E-02 -9.2082605E+00 1.2072737E-01 9.9999996E+24 1.4300000E+03 -9.3707547E-02 0.0000000E+00 -8.4932337E+00 0.0000000E+00 9.9999996E+24 1.5000000E+03 -3.9179351E-02 0.0000000E+00 -3.7248688E+00 0.0000000E+00 9.9999996E+24 1.6000000E+03 -9.8851677E-03 0.0000000E+00 -1.0024587E+00 0.0000000E+00 9.9999996E+24 1.7000000E+03 9.5079672E-03 0.0000000E+00 1.0244696E+00 0.0000000E+00 9.9999996E+24 1.8000000E+03 2.4715576E-02 0.0000000E+00 2.8197184E+00 0.0000000E+00 9.9999996E+24 1.9000000E+03 3.7703570E-02 0.0000000E+00 4.5404463E+00 0.0000000E+00 9.9999996E+24 2.0000000E+03 4.9428467E-02 0.0000000E+00 6.2656999E+00 0.0000000E+00 9.9999996E+24 2.1000000E+03 6.0429081E-02 0.0000000E+00 8.0431795E+00 0.0000000E+00 2 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 3.3390682E+00 4.0000000E+02 -2.8284783E-02 3.4867945E+00 -3.0983400E-01 2.1908176E+0 4.2196832E+00 5.0000000E+02 -3.4942660E-02 3.4489098E+00 -3.7860614E-01 2.1670139E+01 5.1352148E+00 6.0000000E+02 -4.0042900E-02 3.4008248E+00 -4.2781860E-01 2.1368013E+01 6.0952721E+00 7.0000000E+02 -4.2240817E-02 3.3426938E+00 -4.4358701E-01 2.1002766E+01 7.1085973E+00 8.0000000E+02 -3.9737049E-02 3.2756515E+00 -4.0892455E-01 2.0581526E+01 8.1795788E+00 9.0000000E+02 -3.0523464E-02 3.2026036E+00 -3.0710500E-01 2.0122553E+01 9.3037176E+00 1.0000000E+03 -1.3422978E-02 3.1284924E+00 -1.3192706E-01 1.9656898E+01 1.0468166E+01 1.1000000E+03 1.0546523E-02 3.0585368E+00 1.0133813E-01 1.9217354E+01 1.1660662E+01 1.2000000E+03 3.8356557E-02 2.9953644E+00 3.6094347E-01 1.8820431E+01 1.2876984E+01 1.3000000E+03 6.6786908E-02 2.9384673E+00 6.1654115E-01 1.8462934E+01 1.4120013E+01 1.4000000E+03 9.3580134E-02 2.8859217E+00 8.4843409E-01 1.8132782E+01 1.4246002E+01 1.4100000E+03 9.6109807E-02 2.8808308E+00 8.6983192E-01 1.8100794E+01 1.4372321E+01 1.4200000E+03 9.8607071E-02 2.8757629E+00 8.9086324E-01 1.8068953E+01 1.4435614E+01 1.4250000E+03 9.9844672E-02 2.8732355E+00 9.0125149E-01 1.8053072E+01 1.4448281E+01 1.4260000E+03 1.0009081E-01 2.8727310E+00 9.0331465E-01 1.8049902E+01 1.4460953E+01 1.4270000E+03 1.0033698E-01 2.8722265E+00 9.0537727E-01 1.8046732E+01 1.4473624E+01 1.4280000E+03 1.0058212E-01 2.8717229E+00 9.0743011E-01 1.8043568E+01 1.4486303E+01 1.4290000E+03 1.0082768E-01 2.8712187E+00 9.0948582E-01 1.8040400E+01 1.4498988E+01 1.4300000E+03 1.0107257E-01 2.8707144E+00 9.1153467E-01 1.8037231E+01 1.5395497E+01 1.5000000E+03 1.1726157E-01 2.8358886E+00 1.0447077E+00 1.7818415E+01 1.6709805E+01 1.6000000E+03 1.3682486E-01 2.7870209E+00 1.1979953E+00 1.7511370E+01 1.8068224E+01 1.7000000E+03 1.5135424E-01 2.7385778E+00 1.3021756E+00 1.7206993E+01 1.9473568E+01 1.8000000E+03 1.5982650E-01 2.6904109E+00 1.3508816E+00 1.6904350E+01 2.0924421E+01 1.9000000E+03 1.6107894E-01 2.6429670E+00 1.3374588E+00 1.6606253E+01 2.2413221E+01 2.0000000E+0 1.5391463E-01 2.5972717E+00 1.2558771E+00 1.6319139E+01 2.3925329E+01 2.1000000E+03 1.3731916E-01 2.5547769E+00 1.1021330E+00 1.6052137E+01 = 3 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.6140474E+00 4.0000000E+02 -2.9438451E-02 7.2133226E+00 -6.6711420E-01 4.5322643E+01 2.0168877E+00 5.0000000E+02 -3.7501179E-02 7.2157240E+00 -8.5010922E-01 4.5337734E+01 2.4198287E+00 6.0000000E+02 -4.5750387E-02 7.2170258E+00 -1.0372965E+00 4.5345913E+01 2.8230934E+00 7.0000000E+02 -5.4224186E-02 7.2171283E+00 -1.2294401E+00 4.5346558E+01 3.2268467E+00 8.0000000E+02 -6.2981464E-02 7.2161126E+00 -1.4277953E+00 4.5340176E+01 3.6314509E+00 9.0000000E+02 -7.2098888E-02 7.2136316E+00 -1.6339262E+00 4.5324585E+01 4.0371413E+00 1.0000000E+03 -8.1673540E-02 7.2097082E+00 -1.8499030E+00 4.5299934E+01 4.4446402E+00 1.1000000E+03 -9.1822386E-02 7.2035685E+00 -2.0780029E+00 4.5261356E+01 4.8549180E+00 1.2000000E+03 -1.0269001E-01 7.1943407E+00 -2.3209677E+00 4.5203377E+01 5.2694612E+00 1.3000000E+03 -1.1444992E-01 7.1807332E+00 -2.5818686E+00 4.5117878E+01 5.6908121E+00 1.4000000E+03 -1.2730074E-01 7.1605339E+00 -2.8636916E+00 4.4990963E+01 5.7334146E+00 1.4100000E+0 -1.2865296E-01 7.1580939E+00 -2.8931241E+00 4.4975632E+01 5.7761230E+00 1.4200000E+03 -1.3001817E-01 7.1555586E+00 -2.9227891E+00 4.4959702E+01 5.7975187E+00 1.4250000E+03 -1.3070567E-01 7.1542540E+00 -2.9377081E+00 4.4951504E+01 5.8018012E+00 1.4260000E+03 -1.3084354E-01 7.1539903E+00 -2.9406986E+00 4.4949848E+01 5.8060846E+00 1.4270000E+03 -1.3098158E-01 7.1537251E+00 -2.9436920E+00 4.4948181E+01 5.8103690E+00 1.4280000E+0 -1.3111976E-01 7.1534591E+00 -2.9466877E+00 4.4946510E+01 5.8146553E+00 1.4290000E+03 -1.3125803E-01 7.1531920E+00 -2.9496851E+00 4.4944832E+01 5.8189421E+00 1.4300000E+03 -1.3139643E-01 7.1529241E+00 -2.9526846E+00 4.4943150E+01 6.1222348E+00 1.5000000E+03 -1.4142025E-01 7.1313686E+00 -3.1683588E+00 4.4807713E+01 6.5698667E+00 1.6000000E+03 -1.5686841E-01 7.0885115E+00 -3.4933367E+00 4.4538433E+01 7.0430598E+00 1.7000000E+03 -1.7326842E-01 7.0255318E+00 -3.8242695E+00 4.4142719E+01 7.5554371E+00 1.8000000E+03 -1.8911731E-01 6.9343309E+00 -4.1198912E+00 4.3569687E+01 8.1226606E+00 1.9000000E+03 -2.0079058E-01 6.8084288E+00 -4.2947721E+00 4.2778622E+01 8.7541361E+00 2.0000000E+03 -2.0251060E-01 6.6497960E+00 -4.2306390E+00 4.1781902E+01 9.4459066E+00 2.1000000E+03 -1.8885358E-01 6.4709387E+00 -3.8392146E+00 4.0658108E+01 1

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Flutter Analysis Sample Problems

POINT = 4 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.9974 1.0025905E+00 4.0000000E+02 -1.3031543E-02 1.1612561E+01 -4.7541595E-01 7.2963875E+01 0.7979 1.2532856E+00 5.0000000E+02 -1.6632434E-02 1.1612122E+01 -6.0676044E-01 7.2961113E+01 0.6652 1.5034189E+00 6.0000000E+02 -2.0310717E-02 1.1616167E+01 -7.4120438E-01 7.2986534E+01 0.5704 1.7530142E+00 7.0000000E+02 -2.4091350E-02 1.1622624E+01 -8.7966079E-01 7.3027107E+01 0.4995 2.0020337E+00 8.0000000E+02 -2.8009992E-02 1.1630817E+01 -1.0234653E+00 7.3078583E+01 0.4444 2.2503262E+00 9.0000000E+02 -3.2109644E-02 1.1640956E+01 -1.1742865E+00 7.3142288E+01 0.4004 2.4975274E+00 1.0000000E+03 -3.6435921E-02 1.1654171E+01 -1.3340160E+00 7.3225319E+01 0.3646 2.7427378E+00 1.1000000E+03 -4.1034564E-02 1.1673470E+01 -1.5048724E+00 7.3346581E+01 0.3351 2.9842904E+00 1.2000000E+03 -4.5869853E-02 1.1703933E+01 -1.6865882E+00 7.3537979E+01 0.3108 3.2177768E+00 1.3000000E+03 -5.0232433E-02 1.1759235E+01 -1.8557231E+00 7.3885460E+01 0.2913 3.4329169E+00 1.4000000E+03 -4.6051182E-02 1.1870154E+01 -1.7173034E+00 7.4582382E+01 0.2895 3.4541233E+00 1.4100000E+03 -4.4069543E-02 1.1881545E+01 -1.6449828E+00 7.4653954E+01 0.2877 3.4756999E+00 1.4200000E+03 -4.1757211E-02 1.1891529E+01 -1.5599802E+00 7.4716682E+01 0.2868 3.4866982E+00 1.4250000E+03 -4.0513523E-02 1.1895758E+01 -1.5140563E+00 7.4743256E+01 0.2866 3.4889159E+00 1.4260000E+03 -4.0259469E-02 1.1896540E+01 -1.5046608E+00 7.4748169E+01 0.2864 3.4911397E+00 1.4270000E+03 -4.0003765E-02 1.1897299E+01 -1.4951994E+00 7.4752937E+01 0.2863 3.4933696E+00 1.4280000E+03 -3.9746653E-02 1.1898036E+01 -1.4856815E+00 7.4757568E+01 0.2861 3.4956057E+00 1.4290000E+03 -3.9488193E-02 1.1898751E+01 -1.4761094E+00 7.4762062E+01 0.2859 3.4978483E+00 1.4300000E+03 -3.9228391E-02 1.1899445E+01 -1.4664831E+00 7.4766418E+01 0.2693 3.7128718E+00 1.5000000E+03 -1.8196461E-01 1.1759069E+01 -6.7221742E+00 7.3884415E+01 0.2447 4.0859079E+00 1.6000000E+03 -2.0235516E-01 1.1397853E+01 -7.2458143E+00 7.1614822E+01 0.2231 4.4824362E+00 1.7000000E+03 -2.0488223E-01 1.1038916E+01 -7.1052704E+00 6.9359558E+01 0.2037 4.9082031E+00 1.8000000E+03 -1.8748665E-01 1.0674354E+01 -6.2872663E+00 6.7068947E+01 0.1864 5.3653989E+00 1.9000000E+03 -1.5004404E-01 1.0307261E+01 -4.8586082E+00 6.4762428E+01 0.1708 5.8532186E+00 2.0000000E+03 -1.0008541E-01 9.9455061E+00 -3.1271415E+00 6.2489460E+01 0.1572 6.3628674E+00 2.1000000E+03 -4.9355671E-02 9.6063423E+00 -1.4895154E+00 6.0358429E+01 POINT = 5 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.2569 7.9562515E-01 4.0000000E+02 -3.9658464E-02 1.4633328E+01 -1.8231772E+00 9.1943916E+01 0.9983 1.0017462E+00 5.0000000E+02 -5.0275218E-02 1.4527937E+01 -2.2946043E+00 9.1281723E+01 0.8247 1.2124908E+00 6.0000000E+02 -6.0894571E-02 1.4403379E+01 -2.7554522E+00 9.0499107E+01 0.6997 1.4292080E+00 7.0000000E+02 -7.1599685E-02 1.4255886E+01 -3.2066770E+00 8.9572380E+01 0.6048 1.6535462E+00 8.0000000E+02 -8.2352780E-02 1.4082030E+01 -3.6432874E+00 8.8480011E+01 0.5298 1.8875158E+00 9.0000000E+02 -9.3020976E-02 1.3878532E+01 -4.0557795E+00 8.7201393E+01 0.4687 2.1335819E+00 1.0000000E+03 -1.0342184E-01 1.3642134E+01 -4.4324565E+00 8.5716057E+01 0.4175 2.3950694E+00 1.1000000E+03 -1.1338449E-01 1.3367993E+01 -4.7617850E+00 8.3993584E+01 0.3736 2.6767573E+00 1.2000000E+03 -1.2273487E-01 1.3048598E+01 -5.0313172E+00 8.1986763E+01 0.3348 2.9869835E+00 1.3000000E+03 -1.3172621E-01 1.2667829E+01 -5.2423286E+00 7.9594315E+01 0.2992 3.3424144E+00 1.4000000E+03 -1.4796728E-01 1.2191563E+01 -5.6672840E+00 7.6601852E+01 0.2958 3.3802135E+00 1.4100000E+03 -1.5111682E-01 1.2141340E+01 -5.7640710E+00 7.6286293E+01 0.2926 3.4179220E+00 1.4200000E+03 -1.5454833E-01 1.2092547E+01 -5.8712697E+00 7.5979721E+01 0.2910 3.4367015E+00 1.4250000E+03 -1.5633802E-01 1.2068816E+01 -5.9276037E+00 7.5830612E+01 0.2907 3.4404502E+00 1.4260000E+03 -1.5670007E-01 1.2064127E+01 -5.9390225E+00 7.5801147E+01 0.2903 3.4441965E+00 1.4270000E+03 -1.5706323E-01 1.2059456E+01 -5.9504819E+00 7.5771797E+01 0.2900 3.4479396E+00 1.4280000E+03 -1.5742736E-01 1.2054804E+01 -5.9619761E+00 7.5742569E+01 0.2897 3.4516809E+00 1.4290000E+03 -1.5779237E-01 1.2050172E+01 -5.9735036E+00 7.5713463E+01 0.2894 3.4554188E+00 1.4300000E+03 -1.5815827E-01 1.2045560E+01 -5.9850640E+00 7.5684486E+01 0.2726 3.6680081E+00 1.5000000E+03 -2.1511095E-02 1.1902894E+01 -8.0438685E-01 7.4788094E+01 0.2540 3.9365799E+00 1.6000000E+03 -4.9833781E-03 1.1830213E+01 -1.8521078E-01 7.4331421E+01 0.2370 4.2191119E+00 1.7000000E+03 1.6927561E-03 1.1727880E+01 6.2368281E-02 7.3688446E+01 0.2218 4.5080471E+00 1.8000000E+03 -6.8890274E-04 1.1621861E+01 -2.5152635E-02 7.3022308E+01 0.2086 4.7933521E+00 1.9000000E+03 -1.1143429E-02 1.1537346E+01 -4.0390077E-01 7.2491287E+01 0.1974 5.0647664E+00 2.0000000E+03 -2.5090471E-02 1.1493762E+01 -9.0598476E-01 7.2217438E+01 0.1878 5.3240671E+00 2.1000000E+03 -3.7430566E-02 1.1480675E+01 -1.3500308E+00 7.2135208E+01 COMPLEX EIGENVALUE = -1.319271E-01, 1.965690E+01 C O M P L E X E I G E N V E C T O R NO. 2 (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 0.0 0.0 2.508526E-02 0.0 0.0 0.0 0.0 0.0 1.613183E-02 0.0 0.0 0.0 2 G 0.0 0.0 1.031495E-01 0.0 0.0 0.0 0.0 0.0 -3.391594E-02 0.0 0.0 0.0 3 G 0.0 0.0 1.162375E-01 0.0 0.0 0.0 0.0 0.0 2.545503E-02 0.0 0.0 0.0 4 G 0.0 0.0 2.549485E-01 0.0 0.0 0.0 0.0 0.0 -6.140902E-02 0.0 0.0 0.0 5 G 0.0 0.0 3.571462E-01 0.0 0.0 0.0 0.0 0.0 2.425030E-02 0.0 0.0 0.0 6 G 0.0 0.0 4.976830E-01 0.0 0.0 0.0 0.0 0.0 -5.850959E-02 0.0 0.0 0.0 7 G 0.0 0.0 6.736954E-01 0.0 0.0 0.0 0.0 0.0 6.928696E-03 0.0 0.0 0.0 8 G 0.0 0.0 8.139030E-01 0.0 0.0 0.0 0.0 0.0 -6.776229E-02 0.0 0.0 0.0 9 G 0.0 0.0 9.668907E-01 0.0 0.0 0.0 0.0 0.0 -1.876364E-02 0.0 0.0 0.0 10 G 0.0 0.0 1.097138E+00 0.0 0.0 0.0 0.0 0.0 -8.202810E-02 0.0 0.0 0.0 11 G 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 COMPLEX EIGENVALUE = 1.013381E-01, 1.921735E+01 C O M P L E X E I G E N V E C T O R NO. 7 (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 0.0 0.0 3.087929E-02 0.0 0.0 0.0 0.0 0.0 1.569297E-02 0.0 0.0 0.0 2 G 0.0 0.0 8.654780E-02 0.0 0.0 0.0 0.0 0.0 -3.018055E-02 0.0 0.0 0.0 3 G 0.0 0.0 1.257552E-01 0.0 0.0 0.0

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0.0 0.0 2.450372E-02 0.0 0.0 0.0 2.244809E-01 0.0 0.0 0.0 -5.494687E-02 0.0 5 G 0.0 0.0 3.649569E-01 0.0 0.0 0.0 2.340965E-02 0.0 6 G 0.0 0.0 4.677418E-01 0.0 0.0 0.0 -5.238892E-02 0.0 7 G 0.0 0.0 6.779492E-01 0.0 0.0 0.0 5.586534E-03 0.0 8 G 0.0 0.0 7.839131E-01 0.0 0.0 0.0 -6.282084E-02 0.0 9 G 0.0 0.0 9.677300E-01 0.0 0.0 0.0 -2.040747E-02 0.0 10 G 0.0 0.0 1.068483E+00 0.0 0.0 0.0 -7.822070E-02 0.0 11 G 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 COMPLEX EIGENVALUE = -1.002459E+00, 0.000000E+00 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 1 G 0.0 0.0 4.771255E-02 0.0 0.0 0.0 0.0 0.0 2 G 0.0 0.0 3.074158E-02 0.0 0.0 0.0 0.0 0.0 3 G 0.0 0.0 1.514307E-01 0.0 0.0 0.0 0.0 0.0 4 G 0.0 0.0 1.231133E-01 0.0 0.0 0.0 0.0 0.0 5 G 0.0 0.0 3.785414E-01 0.0 0.0 0.0 0.0 0.0 6 G 0.0 0.0 3.449427E-01 0.0 0.0 0.0 0.0 0.0 7 G 0.0 0.0 7.000237E-01 0.0 0.0 0.0 0.0 0.0 8 G 0.0 0.0 6.606572E-01 0.0 0.0 0.0 0.0 0.0 9 G 0.0 0.0 1.028760E+00 0.0 0.0 0.0 0.0 0.0 10 G 0.0 0.0 9.876114E-01 0.0 0.0 0.0 0.0 0.0 11 G 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 COMPLEX EIGENVALUE = 1.024470E+00, 0.000000E+00 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 1 G 0.0 0.0 4.814341E-02 0.0 0.0 0.0 0.0 0.0 2 G 0.0 0.0 3.111237E-02 0.0 0.0 0.0 0.0 0.0 3 G 0.0 0.0 1.520088E-01 0.0 0.0 0.0 0.0 0.0 4 G 0.0 0.0 1.237039E-01 0.0 0.0 0.0 0.0 0.0 5 G 0.0 0.0 3.792227E-01 0.0 0.0 0.0 0.0 0.0 6 G 0.0 0.0 3.457828E-01 0.0 0.0 0.0 0.0 0.0 7 G 0.0 0.0 6.995916E-01 0.0 0.0 0.0 0.0 0.0 8 G 0.0 0.0 6.606771E-01 0.0 0.0 0.0 0.0 0.0 9 G 0.0 0.0 1.026566E+00 0.0 0.0 0.0 0.0 0.0 10 G 0.0 0.0 9.861466E-01 0.0 0.0 0.0 0.0 0.0 11 G 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4

G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 12 R2

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

18 R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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Listing 7-7. Output for Jet Transport Wing by Lifting Surface Theory

7.4 Flutter and Divergence Analysis of Jet Transport Wing by Strip Theory (Example HA145C) This example is the same BAH wing considered in Example HA145B with only the aerodynamic theory and the flutter solution method changed. Aerodynamic Strip Theory is used and the solution is obtained by the K-Flutter method. The spanwise divisions shown in Figure 7-3 are also used in the Strip Theory, and the analysis again assumes sea-level density.

Strip Theory Aerodynamics CAERO4 4000 is the Bulk Data entry for the Strip Theory aerodynamics. It invokes property entry PAERO4 that lists the aerodynamic and geometrical characteristics of each strip. The basic coordinate system is specified and nonuniform strip divisions are given on AEFACT 77 for the six strips; the continuation entry tabulates the planform geometry at the root and tip chords. PAERO4 4001 provides CLA = 0 for no Prandtl-Glauert compressibility correction, LCLA = 0 for the theoretical two-dimensional lift curve slope on each strip to be clα = 2π, CIRC = 0 for the exact circulation function of Theodorsen, and LCIRC = 0 since no approximation is used for the circulation function. The remaining required data provide any control surface geometry, and, since there is no aileron, DOCi = CAOCi = GAPOCi = 0.0 for all six strips. SPLlNE2 100 is the linear spline along the 35%-chord elastic axis that connects all 11 structural grid points to the five strips (IDs 4001 through 4005) outboard of the fuselage side, i.e., the inboard strip (ID 4000), which is assumed to represent the fuselage, does not move. The aerodynamics for the flutter analysis are specified on MKAERO1 entries for a Mach number m = 0.0 and reduced frequencies k from 0.0001 (to obtain the divergence speed) up to 1.5; the 11 values of k require two separate MKAERO1 entries. FLUTTER 30 specifies the K-method of flutter analysis with a density ratio of 1.0 on FLFACT 1, a Mach number m = 0.0 on FLFACT 2, and a series of reduced frequencies decreasing from k = 0.20 to k = 0.0001; the two small values of k determine the divergence speed, and the remaining values determine the flutter speed. The K-method also requires an EIGC entry to specify the complex eigenvalue method. EIGC 20 requests the Upper Hessenberg method to determine all 10 eigenvalues with the modal participation factors (eigenvectors of the modal solution) normalized on their maximum component. The ENDDATA entry completes the Bulk Data Section. The Case Control Section includes the title, subtitle, and label. ECHO = BOTH prints both the annotated and sorted input files. SPC = 1 refers to the SPC1 1 entry, which constrains the fuselage additionally in plunge, roll, and pitch. SDAMP = 2000 refers to the damping entry TABDMP1. The vibration control METHOD = 10 refers to ElGR 10. The complex eigenvalue control CMETHOD = 20 refers to EIGC 20, and the flutter control FMETHOD = 30 refers to FLUTTER 30, which specifies the K-method. SDISP(PLOT) = ALL and DISP(PLOT) = ALL save the eigenvector and displacement data for plotting but do not print them. Note that SDISP = ALL and DISP = ALL will lead to an extensive amount of printed output. The remaining Case Control entries set up the V-g and V-f curves for plotting. The command BEGIN BULK ends the Case Control Section. The Executive Control Section begins with the identification ID NXN, HA145C. Time 5 limits CPU time to 5.0 minutes. SOL 145 calls for the Aerodynamic Flutter DMAP solution sequence, and CEND concludes the Executive Control Section. The input data echo is shown in Listing

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7-8 followed by the sorted Bulk Data entries in Listing 7-9. Selected output data are shown in Listing 7-10 and in Figure 7-6. The primary results of the analysis are discussed as follows. The K-flutter analysis shows the same number of instabilities as the PK-flutter analysis of Example HA145B , but at somewhat different speeds because of the significant differences between Lifting Surface Theory and Strip Theory. The critical flutter speed in the V-g curve of Figure 7-6 or in the Flutter Summary for Point 6 (k = 0.090) is 1152 f/s = 786 mph. This speed is 9.1% higher than that obtained by Lifting Surface Theory. A comparison of the Lifting Surface and Strip Theory damping results is shown in Figure 7-7 for the critical second (torsion) flutter mode. The differences in frequencies are small whereas the differences in damping are substantial. The divergence speeds are found from the K-flutter analysis at very low reduced frequencies. From the Flutter Summary, Point 12 (k = 0.0010) shows the lowest divergence speed to be 1421.4 ft/s (with an artificial structural damping of g = −0.0250), and Point 13 (k = 0.0001) shows the lowest divergence speed to be 1419.8 ft/s (with g = −0.00303). Point 13, corresponding to a very low value of k, is the more accurate result and agrees almost exactly with the value of 1419.9 ft/s obtained by regarding divergence as a static instability (eigenvalue) problem in Bisplinghoff, Ashley, and Halfman (1955). The critical speeds of the higher divergence modes are also estimated from Point 13. The second speed is 2873 ft/s and the third speed is 4486 ft/s, approximately in the ratio 1:2:3 as theoretically predicted for a uniform wing by Strip Theory. The divergence speed of 1420 ft/s from Strip Theory is 14.0% lower than the 1651 ft/s obtained from Lifting Surface Theory in Example HA145B. It is important to note that in this example Strip Theory is conservative for the prediction of divergence, but unconservative for the prediction of flutter.

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Figure 7-6. BAH Wing V-g and V-f Curves for First and Second Modes

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Figure 7-7. Stability Curves for Critical Torsion Mode of BAH Wing N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA145C $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145C $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ $ CANTILEVERED WING WITH TEN BEAM $ $ ELEMENTS AND DUMBBELL MASSES $ $ $ $ SOLUTION K METHOD OF FLUTTER ANALYSIS $ $ USING STRIP THEORY WITH THE EXACT $ $ THEODORSEN FUNCTION AT MACH NO. 0.0 $ $ $ $ OUTPUT TABULATED MODAL DEFLECTIONS PLUS $ $ X-Y PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ TIME IN CPU MINUTES SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145C: BAH JET TRANSPORT WING FLUTTER ANALYSIS PAGE 2 CANTILEVERED, EXACT STRIP THEORY AT MACH NO. 0.0 K-FLUTTER METHOD C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145C: BAH JET TRANSPORT WING FLUTTER ANALYSIS 2 SUBTI = CANTILEVERED, EXACT STRIP THEORY AT MACH NO. 0.0 3 LABEL = K-FLUTTER METHOD 4 ECHO = BOTH 5 SPC = 1 $ FUSELAGE CONSTRAINT 6 SDAMP = 2000 $ STRUCTURAL DAMPING 7 METHOD = 10 $ MODIFIED GIVENS FOR VIBRATION ANALYSIS 8 SVEC = ALL $ PRINT VIBRATION MODES 9 CMETHOD = 20 $ HESSENBERG METHOD FOR FLUTTER ANALYSIS 10 FMETHOD = 30 $ K-FLUTTER METHOD 11 $---BE CAREFUL WITH EIGENVECTOR TABULATIONS: THE SDISP=ALL COMMAND PRINTS 12 $ ONE MODAL EIGENVECTOR PER PAGE FOR EACH EIGENVALUE AND FOR ALL THE 13 $ VALUES OF REDUCED FREQUENCY; DISP=ALL PRINTS THE CORRESPONDING 14 $ GRID DISPLACEMENTS ONE SET PER PAGE. SDISP(PLOT)=ALL AND DISP(PLOT) 15 $ =ALL SAVES THEM BUT DOES NOT PRINT THEM. 16 SDISP(PLOT) = ALL $ PLOT AND SAVE ALL MODAL PARTICIPATION FACTORS 17 SET 10 = 1 THRU 5000 18 DISP(PLOT) = 10 $ PLOT AND SAVE STRUCTURAL DISPLACEMENTS IN SET 10 19 OUTPUT(XYOUT) 20 CSCALE 2.0

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21 PLOTTER NASTRAN 22 CURVELINESYMBOL = -6 23 YTTITLE = DAMPING G 24 YBTITLE =FREQUENCY F HZ 25 XTITLE = VELOCITY V (FT/SEC) 26 XMIN = 0. 27 XMAX = 2500. 28 YTMIN = -.6 29 YTMAX = +.1 30 YBMIN = 0. 31 YBMAX = 15. 32 XTGRID LINES = YES 33 XBGRID LINES = YES 34 YTGRID LINES = YES 35 YBGRID LINES = YES 36 UPPER TICS = -1 37 TRIGHT TICS = -1 38 BRIGHT TICS = -1 39 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 4(G,F) 5(G,F) 6(G,F) 7(G,F), 40 8(G,F) 9(G,F) 10(G,F) 11(G,F) 12(G,F) 13(G,F) 14(G,F), 41 15(G,F) 16(G,F) 17(G,F) 18(G,F) 19(G,F) 42 CURVELINESYMBOL = -6 43 YTTITLE = DAMPING G 44 YBTITLE =FREQUENCY F HZ 45 XTITLE = VELOCITY V (FT/SEC) 46 XMIN = 0. 47 XMAX = 1600. 48 YTMIN = -1.7 49 YTMAX = +.1 50 YBMIN = 0. 51 YBMAX = 2.5 52 XTGRID LINES = YES 53 XBGRID LINES = YES 54 YTGRID LINES = YES 55 YBGRID LINES = YES 56 UPPER TICS = -1 57 TRIGHT TICS = -1 58 BRIGHT TICS = -1 59 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 4(G,F) 5(G,F) 6(G,F) 7(G,F), 60 8(G,F) 9(G,F) 10(G,F) 11(G,F) 12(G,F) 13(G,F) 14(G,F), 61 15(G,F) 16(G,F) 17(G,F) 18(G,F) 19(G,F) 62 BEGIN BULK EXAMPLE HA145C: BAH JET TRANSPORT WING FLUTTER ANALYSIS PAGE CANTILEVERED, EXACT STRIP THEORY AT MACH NO. 0.0 K-FLUTTER METHOD I N P U T B U L K D A T A D E C K E C H O .

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4

1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ INCLUDE BAH_STRUCT.DAT INCLUDE BAH_MASS.DAT $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ THE SPC ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS $ $ AND ENFORCED DISPLACEMENTS. IT LISTS THE ID, GRID POINT $ $ NO., CONSTRAINED DOFS AND VALUE OF AN ENFORCED DISPLACE$ $ MENT. $ $ $ $ SID G C D SPC 1 11 345 $ $ $ * * STRUCTURAL DAMPING * * $ $ $ $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUSION $ $ OF STRUCTURAL DAMPING IN EQUATIONS OF MOTION (SEE HANDBOOK $ $ FOR DYNAMIC ANALYSIS, SECT. 3.2.2). IF SET TO -1, MODAL $ $ DAMPING IS PUT INTO COMPLEX STIFFNESS MATRIX AS STRUCTURAL $ $ DAMPING. $ $ $ $ N V1 V2 PARAM KDAMP -1 $ $ $ THE TABDMP1 ENTRY DEFINES MODAL DAMPING AS A TABULAR $ $ FUNCTION OF FREQUENCY. THE DAMPING LEVELS ARE LINEAR $ $ BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE EXTRAP$ $ OLATED OUTSIDE THE TABULATED FREQUENCY RANGE. $ $ $ $ ID TYPE +TDP TABDMP1 2000 G +T2000 $ F1 G1 F2 G2 ETC ENDT +T2000 0.0 0.0 10.0 0.0 ENDT $ $ * * * AERODYNAMIC DATA * * * $ $ (LB-IN-SEC SYSTEM) $

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

$

$ * * ELEMENT GEOMETRY * * $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD $ AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=1 INDICATES $ THAT THE MODEL IS MOUNTED WITH A ROOT REFLECTION PLANE; $ SYMXY = 0 INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH $ FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ NOTE THAT THE THE DIMENSIONS OF THE REFERENCE DENSITY OF $ AIR ARE SNAILS PER CUBIC INCH, NOT POUNDS PER CUBIC INCH. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 131.232 1.1468-7 1 $ $ THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM $ FLAGGED BY THE AERO ENTRY. LISTED ARE THE ORIGIN, A $ POINT ALONG THE Z AXIS AND A POINT IN THE X-Z PLANE, $ ALL IN THE RID COORDINATE SYSTEM. NOTE THAT IN THIS $ COORDINATE SYSTEM THE UPSIDE DOWN AND BACKWARDS STRUC$ TURE WILL FLY UPSIDE UP AND FORWARD. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0. 0. 0. 0. 0. -1. $ C1 C2 C3 +C1 -1. 0. 0. THE CAERO4 ENTRY IS USED TO SPECIFY STRIP THEORY AERO. $ $ PID IDENTIFIES A PAERO4 ENTRY WHERE THE ELEMENT PROPERTIES $ ARE GIVEN, CP DEFAULTS TO THE BASIC COORDINATE SYSTEM, $ NSPAN WOULD BE THE NUMBER OF EQUAL SPAN STRIPS, HOWEVER $ LSPAN SPECIFIES THE AEFACT ENTRY WHERE THE UNEQUAL STRIP $ WIDTHS ARE DEFINED. THE CONTINUATION ENTRY DEFINES THE $ INBD AND OUTBD LEADING EDGE POINTS AND THE CHORDS. $ EID PID CP NSPAN LSPAN CAERO4 4000 4001 0 0 77 $ X1 Y1 Z1 X12 X4 Y4 Z4 X43 +BAH 78.75 0.0 0.0 225.0 35.0 500.0 0.0 100.0 $ $ THIS AEFACT ENTRY GIVES THE EDGES OF THE STRIPS IN PERCENT $ OF SPAN. $ $ SID D1 D2 D3 ETC AEFACT 77 .0 .09 .276 .454 .636 .826 1.0 $ $ THE PAERO4 ENTRY DEFINES PROPERTIES OF THE STRIP ELEMENTS $ FOR STRIP THEORY AERODYNAMICS. IT OFFERS A WIDE VARIETY $ OF OPTIONS. CLA DEFAULTS TO NO COMPRESSIBILITY CORRECTION $ AND NO CORRECTIONS OF LIFT CURVE SLOPES. LCLA MUST BE ZERO $ WHEN CLA IS ZERO. CIRC=0 SPECIFIES THE EXACT EXPRESSION FOR $ THE THEODORSEN CIRCULATION FUNCTION. DOC, CAOC AND GAPOC $ ARE REQUIRED EVEN THOUGH THE STRIP HAS NO CONTROL SURFACE. $ IN FRACTIONS OF CHORD THESE ARE THE DISTANCE FROM THE QUARTER $ CHORD TO THE HINGE LINE, THE CONTROL SURFACE CHORD AND THE $ GAP BETWEEN THE PRIMARY SURFACE AND THE CONTROL SURFACE. $ CAOC1=0. SPECIFIES NO CONTROL SURFACE. $ PID CLA LCLA CIRC LCIRC DOC1 CAOC1 GAPOC1 PAERO4 4001 0 0 0 0 0.0 0.0 0.0 +PA41 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 +PA42 0.0 0.0 0.0 0.0 0.0 0.0 0.0 $ $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * BEAM SPLINE FIT ON THE WING * $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT$ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR $ ATTACHMENT AND TORSIONAL FLEXIBILITIES. DTHX AND DTHY $ ARE ROTATIONAL ATTACHMENT FLEXIBILITIES. CID IDENTIFIES $ THE SPLINE AXIS. $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 100 4000 4001 4005 14 .0 1.0 0 $ DTHX DTHY +SP100 -1.0 -1.0 $ $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 G5 G6 SET1 14 1 THRU 11 $ $ * * * SOLUTION SPECIFICATIONS * * * $ * * AERODYNAMIC CONDITIONS * * $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ TO GENERATE GENERALIZED AERO FORCE MATRICES. IF MORE THAN $ EIGHT MACH NO.S OR REDUCED FREQUENCIES ARE REQUIRED A SECOND $ MKAERO1 ENTRY IS NECESSARY. $

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +C1

$ $ $ $ $ $ +BAH $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +PA41 +PA42 $ $ $ $ $ $ $ $ $ $ $ $ $ +SP100 $ $ $ $ $ $ $ $ $ $ $ $ $ $

7-35

Chapter 7

Flutter Analysis Sample Problems

$ MKAERO1 $ +MK1 MKAERO1 +MK2 $ $ $ $ $ $ $

7-36

M1 0.0 K1 0.0001 0.0 0.50

M2

M3

ETC

K2 0.02

K3 0.04

K4 0.06

1.0

1.5

+MK1 ETC 0.08

0.10

0.12

0.20 +MK2

* * SOLUTION PARAMETERS * * THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE CRITERION FOR CONVERGENCE (DEFAULT IS 10-3). $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 30 K 1 2 3 L 6 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NO.S $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 1. FLFACT 2 .0 FLFACT 3 0.20 0.16 0.12 0.11 0.10 0.09 0.08 +FLF3 0.07 0.06 0.05 0.04 0.001 0.0001 $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. $ $ N V1 V2 PARAM LMODES 10 $ $ THE PARAM,VREF,C ENTRY SPECIFIES A CONVERSION FACTOR TO BE $ USED TO CONVERT THE DIMENSIONS OF THE OUTPUT VELOCITIES BY $ DIVIDING BY C, IN THIS CASE BY 12.0 IN/FT TO PRINT VEL$ OCITIES IN FT/SEC RATHER THAN IN/SEC. $ $ N V1 V2 PARAM VREF 12.0 $ $ * * EIGENSOLUTION PARAMETERS * * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM; IN THIS CASE THE $ MODIFIED GIVENS METHOD. THREE MODES ARE DESIRED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 ND EIGR 10 MGIV 10 $ NORM G C +EIGR MAX $ $ THE EIGC ENTRY DEFINES DATA NEEDED TO PERFORM A COMPLEX $ EIGENVALUE ANALYSIS. THIS ONE SPECIFIES THAT THE UPPER $ HESSENBERG METHOD BE USED TO EXTRACT COMPLEX EIGENVALUES $ AND THAT THE EIGENVECTORS BE NORMALIZED ON THEIR MAXIMUM $ COMPONENTS. IF THEY WERE TO HAVE BEEN NORMALIZED ON A $ POINT THE GRID AND DOF COMPONENT WOULD HAVE BEEN REQUIRED. $ THE CONVERGENCE CRITERION DEFAULTS TO 1.0-4 FOR THE INVERSE $ POWER METHOD, TO 1.0-11 FOR THE DETERMINANT METHOD AND TO $ 1.0-15 FOR THE HESSENBERG METHOD. WHEN THE HESSENBERG $ METHOD IS SPECIFIED ALL ENTRIES ON THE CONTINUATION ENTRY $ EXCEPT THE NUMBER OF DESIRED ROOTS (NDL) ARE IGNORED. $ $ SID METHOD NORM G C E EIGC 20 HESS MAX $ ALPHAA1 OMEGAA1 ALPHAB1 OMEGAB1 L1 NEL NDL +EC 10 ENDDATA INPUT BULK DATA CARD COUNT = 359

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO +FLF3 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +EIGR $ $ $ $ $ $ $ $ $ $ $ $ $ +EC

Flutter Analysis Sample Problems

Listing 7-8. Input for Jet Transport Wing by Strip Theory EXAMPLE HA145C: BAH JET TRANSPORT WING FLUTTER ANALYSIS PAGE 12 CANTILEVERED, EXACT STRIP THEORY AT MACH NO. 0.0 K-FLUTTER METHOD S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 77 .0 .09 .276 .454 .636 .826 1.0 2AERO 1 131.232 1.1468-71 3CAERO4 4000 4001 0 0 77 +BAH 4+BAH 78.75 0.0 0.0 225.0 35.0 500.0 0.0 100.0 5CMASS2 121 5248.7 1 3 6CMASS2 122 134.9 1 3 2 3 7CMASS2 123 790.3 2 3 8CMASS2 341 9727. 3 3 9CMASS2 342 11005. 3 3 4 3 10CMASS2 343 473. 4 3 11CMASS2 561 3253.6 5 3 12CMASS2 562 -139.7 5 3 6 3 13CMASS2 563 946.3 6 3 14CMASS2 781 2617.8 7 3 15CMASS2 782 21. 7 3 8 3 16CMASS2 783 782.3 8 3 17CMASS2 9101 494.8 9 3 18CMASS2 9102 -7.3 9 3 10 3 19CMASS2 9103 185.2 10 3 20CONM1 1 11 +51 21+51 17400. 4.37+7 +52 22+52 4.35+09 23CORD2R 1 0. 0. 0. 0. 0. -1. +C1 24+C1 -1. 0. 0. 25EIGC 20 HESS MAX +EC 26+EC 10 27EIGR 10 MGIV 10 +EIGR 28+EIGR MAX 29FLFACT 1 1. DENSITY 30FLFACT 2 .0 MACH NO 31FLFACT 3 0.20 0.16 0.12 0.11 0.10 0.09 0.08 +FLF3 32+FLF3 0.07 0.06 0.05 0.04 0.001 0.0001 33FLUTTER 30 K 1 2 3 L 6 34GENEL 432 1 3 2 3 3 3 +01 35+01 4 3 5 3 6 3 7 3 +02 36+02 8 3 9 3 10 3 +03 37+03 UD 11 3 11 4 11 5 +04 38+04 11 6 +05 39+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 40+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 41+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 42+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 43+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 44+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 45+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 46+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 47+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 48+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 49+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 50+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 51+17 413.0 52GRID 1 20.25 90. 12456 53GRID 2 -81. 90. 12456 54GRID 3 17.85 186. 12456 55GRID 4 -71.4 186. 12456 56GRID 5 15.8 268. 12456 57GRID 6 -63.2 268. 12456 58GRID 7 13.3 368. 12456 59GRID 8 -53.2 368. 12456 60GRID 9 11.05 458. 12456 61GRID 10 -44.2 458. 12456 62GRID 11 0.0 0. 126 63MKAERO1 0.0 +MK1 64+MK1 0.0001 0.02 0.04 0.06 0.08 0.10 0.12 0.20 65MKAERO1 0.0 +MK2 66+MK2 0.50 1.0 1.5 67PAERO4 4001 0 0 0 0 0.0 0.0 0.0 +PA41 68+PA41 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 +PA42 69+PA42 0.0 0.0 0.0 0.0 0.0 0.0 0.0 70PARAM GRDPNT 11 71PARAM KDAMP -1 72PARAM LMODES 10 73PARAM VREF 12.0 74PARAM WTMASS .0025907 75SET1 14 1 THRU 11 76SPC 1 11 345 77SPLINE2 100 4000 4001 4005 14 .0 1.0 0 +SP100 78+SP100 -1.0 -1.0 79TABDMP1 2000 +T2000 80+T2000 0.0 0.0 10.0 0.0 ENDT

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 7

Flutter Analysis Sample Problems

ENDDATA TOTAL COUNT=

81

Listing 7-9. Sorted Bulk Data Entries for Jet Transport Wing by Strip Theory EXAMPLE HA145C: BAH JET TRANSPORT WING FLUTTER ANALYSIS PAGE 15 CANTILEVERED, EXACT STRIP THEORY AT MACH NO. 0.0 K-FLUTTER METHOD O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 11 M O * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 4.191900E+04 5.128960E+06 -1.642074E+05 0.000000E+00 * * 0.000000E+00 0.000000E+00 5.128960E+06 1.350243E+09 -2.381847E+07 0.000000E+00 * * 0.000000E+00 0.000000E+00 -1.642074E+05 -2.381847E+07 4.458782E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Y 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Z 4.191900E+04 3.917256E+00 1.223541E+02 0.000000E+00 I(S) * 7.226942E+08 3.727022E+06 0.000000E+00 * * 3.727022E+06 4.458139E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 * I(Q) * 4.458143E+09 * * 7.226906E+08 * * 0.000000E+00 * Q * 9.977437E-04 9.999995E-01 0.000000E+00 * * -9.999995E-01 9.977437E-04 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 1 1.637768E+02 1.279753E+01 2.036790E+00 8.160930E+00 1.336571E+03 2 2 4.982469E+02 2.232145E+01 3.552568E+00 5.525822E+01 2.753224E+04 3 3 2.092550E+03 4.574440E+01 7.280447E+00 7.079897E+00 1.481504E+04 4 4 5.402874E+03 7.350424E+01 1.169856E+01 8.652712E+00 4.674951E+04 5 5 8.742090E+03 9.349915E+01 1.488085E+01 4.002357E+00 3.498896E+04 6 8 1.766007E+04 1.328912E+02 2.115029E+01 3.883447E+00 6.858195E+04 7 6 2.398459E+04 1.548696E+02 2.464826E+01 3.597392E+00 8.628196E+04 8 10 4.211864E+04 2.052283E+02 3.266309E+01 3.142601E+00 1.323621E+05 9 9 6.020812E+04 2.453734E+02 3.905239E+01 1.016253E+00 6.118669E+04 10 7 9.183204E+04 3.030380E+02 4.823000E+01 8.617019E+00 7.913184E+05 EIGENVALUE = 1.637768E+02 CYCLES = 2.036790E+00 R E A L E I G E N V E C T O R N O . 1 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 4.518474E-02 2 G 4.250221E-02 3 G 1.482559E-01 4 G 1.437271E-01 5 G 3.794104E-01 6 G 3.767312E-01 7 G 6.929946E-01 8 G 6.926546E-01 9 G 9.989689E-01 10 G 1.000000E+00 EIGENVALUE = 4.982469E+02 CYCLES = 3.552568E+00 R E A L E I G E N V E C T O R N O . 2 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G -1.635563E-01 2 G 5.449901E-01 3 G -2.796319E-01 4 G 1.000000E+00 5 G -2.267440E-01 6 G 9.465655E-01 7 G -1.388892E-01 8 G 8.878612E-01 9 G -3.973597E-02 10 G 8.224477E-01 FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.2000 5.0000000E+00 3.5462509E+02 -2.0858943E-01 2.0643868E+00 -4.3215588E+02 4.1881943E+03 0.2000 5.0000000E+00 5.8322821E+02 -5.2674945E-02 3.3951595E+00 -1.8401012E+02 6.9914707E+03 0.2000 5.0000000E+00 1.2520754E+03 -2.3927695E-01 7.2887340E+00 -1.7359998E+03 1.4715182E+04 0.2000 5.0000000E+00 1.7207372E+03 -9.8075986E-02 1.0016965E+01 -1.0065363E+03 2.0574883E+04 0.2000 5.0000000E+00 1.9962714E+03 -1.5567349E-01 1.1620938E+01 -1.8368853E+03 2.3741326E+04 0.2000 5.0000000E+00 3.2160227E+03 -3.7120324E-01 1.8721504E+01 -6.6058672E+03 3.6778152E+04 POINT = 2 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.1600 6.2500000E+00 4.4598291E+02 -2.8613585E-01 2.0769680E+00 -7.2885205E+02 5.1966719E+03 0.1600 6.2500000E+00 7.1119800E+02 -5.5524200E-02 3.3120899E+00 -2.3647678E+02 8.5245313E+03 0.1600 6.2500000E+00 1.5065197E+03 -3.0820668E-01 7.0159483E+00 -2.6319680E+03 1.7475635E+04 0.1600 6.2500000E+00 1.9703566E+03 1.7822902E-03 9.1760645E+00 2.1070440E+01 2.3644250E+04 0.1600 6.2500000E+00 2.4891079E+03 -2.6804453E-01 1.1591919E+01 -3.8329753E+03 2.9104336E+04 0.1600 NX Nastran 6.2500000E+00 3.6857942E+03 -2.0898052E-01 1.7164955E+01 -4.4995903E+03 4.3527434E+04 Aeroelastic Analysis User’s Guide 7-38 POINT = 3 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K

Flutter Analysis Sample Problems

KFREQ 1./KFREQ 0.1200 8.3333340E+00 0.1200 8.3333340E+00 0.1200 8.3333340E+00 0.1200 8.3333340E+00 0.1200 8.3333340E+00 0.1200 8.3333340E+00 POINT = 4 MACH NUMBER = KFREQ 1./KFREQ 0.1100 9.0909090E+00 0.1100 9.0909090E+00 0.1100 9.0909090E+00 0.1100 9.0909090E+00 0.1100 9.0909090E+00 0.1100 9.0909090E+00 POINT = 5 MACH NUMBER = KFREQ 1./KFREQ 0.1000 1.0000000E+01 0.1000 1.0000000E+01 0.1000 1.0000000E+01 0.1000 1.0000000E+01 0.1000 1.0000000E+01 0.1000 1.0000000E+01 POINT = 6 MACH NUMBER = KFREQ 1./KFREQ 0.0900 1.1111111E+01 0.0900 1.1111111E+01 0.0900 1.1111111E+01 0.0900 1.1111111E+01 0.0900 1.1111111E+01 0.0900 1.1111111E+01 POINT = 7 MACH NUMBER = KFREQ 1./KFREQ 0.0800 1.2500000E+01 0.0800 1.2500000E+01 0.0800 1.2500000E+01 0.0800 1.2500000E+01 0.0800 1.2500000E+01 0.0800 1.2500000E+01 POINT = 8 MACH NUMBER = KFREQ 1./KFREQ 0.0700 1.4285714E+01 0.0700 1.4285714E+01 0.0700 1.4285714E+01 0.0700 1.4285714E+01 0.0700 1.4285714E+01 0.0700 1.4285714E+01 POINT = 9 MACH NUMBER = KFREQ 1./KFREQ 0.0600 1.6666668E+01 0.0600 1.6666668E+01 0.0600 1.6666668E+01 0.0600 1.6666668E+01 0.0600 1.6666668E+01 0.0600 1.6666668E+01 POINT = 10 MACH NUMBER = KFREQ 1./KFREQ 0.0500 2.0000000E+01 0.0500 2.0000000E+01 0.0500 2.0000000E+01 0.0500 2.0000000E+01 0.0500 2.0000000E+01 0.0500 2.0000000E+01 POINT = 11 MACH NUMBER = KFREQ 1./KFREQ 0.0400 2.5000000E+01 0.0400 2.5000000E+01 0.0400 2.5000000E+01 0.0400 2.5000000E+01 0.0400 2.5000000E+01 0.0400 2.5000000E+01 POINT = 12 MACH NUMBER = KFREQ 1./KFREQ 0.0010 9.9999994E+02 0.0010 9.9999994E+02 0.0010 9.9999994E+02 0.0010 9.9999994E+02 0.0010 9.9999994E+02 0.0010 9.9999994E+02 POINT = 13 MACH NUMBER = KFREQ 1./KFREQ 0.0001 1.0000000E+04 0.0001 1.0000000E+04 0.0001 1.0000000E+04 0.0001 1.0000000E+04 0.0001 1.0000000E+04 0.0001 1.0000000E+04

VELOCITY DAMPING FREQUENCY COMPLEX 5.9679846E+02 -4.3520951E-01 2.0844936E+00 -1.3976279E+03 9.0675781E+02 -3.9083730E-02 3.1671174E+00 -2.1243411E+02 1.8064015E+03 -2.5740314E-01 6.3093863E+00 -2.6800193E+03 2.4507854E+03 -1.1014013E-04 8.5600863E+00 -1.6195790E+00 3.0537751E+03 -3.9544508E-01 1.0666204E+01 -6.6144561E+03 4.3100171E+03 8.2674377E-02 1.5053997E+01 2.1288835E+03 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 6.4975281E+02 -4.9224150E-01 2.0803311E+00 -1.6744279E+03 9.7453589E+02 -2.8433582E-02 3.1201978E+00 -1.6617331E+02 1.8861669E+03 -2.0553094E-01 6.0389915E+00 -2.2665554E+03 2.6655566E+03 -4.7289357E-02 8.5343847E+00 -7.5525952E+02 3.1571689E+03 -3.9091003E-01 1.0108393E+01 -6.7731387E+03 4.5606655E+03 1.2136488E-01 1.4602006E+01 3.2908052E+03 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 7.1100360E+02 -5.5938053E-01 2.0694907E+00 -2.0106278E+03 1.0549010E+03 -1.4948593E-02 3.0704591E+00 -9.4602501E+01 1.9620721E+03 -1.3831225E-01 5.7109265E+00 -1.6090920E+03 3.1564470E+03 -3.6192262E-01 9.1873474E+00 -6.3452896E+03 3.0453479E+03 -1.1771613E-01 8.8639755E+00 -2.1324919E+03 4.8639575E+03 1.4269884E-01 1.4157331E+01 4.1123213E+03 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 7.8175446E+02 -6.3622713E-01 2.0478799E+00 -2.4087664E+03 1.1524427E+03 -4.5879173E-05 3.0189331E+00 -3.1723872E-01 2.0333984E+03 -6.4582519E-02 5.3266802E+00 -7.8588477E+02 3.2243340E+03 -4.6963042E-01 8.4464483E+00 -8.0164155E+03 3.5496467E+03 -5.9229746E-02 9.2986374E+00 -1.2587098E+03 5.1131235E+03 -2.5528294E-01 1.3394312E+01 -7.5282769E+03 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 8.6312787E+02 -7.2019100E-01 2.0098181E+00 -2.8646790E+03 1.2736655E+03 1.2577129E-02 2.9657667E+00 9.6104836E+01 2.1041970E+03 9.7037219E-03 4.8996830E+00 1.2250404E+02 3.4336306E+03 -5.6644344E-01 7.9953070E+00 -9.7949258E+03 4.0859543E+03 1.0565862E-02 9.5142622E+00 2.5901172E+02 5.1808037E+03 -2.2965454E-01 1.2063651E+01 -6.9128076E+03 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 9.5593756E+02 -8.0510581E-01 1.9476870E+00 -3.3659946E+03 1.4277452E+03 1.4557940E-02 2.9089777E+00 1.2469366E+02 2.1785779E+03 8.7926425E-02 4.4387712E+00 1.1438076E+03 3.6923875E+03 -6.2787712E-01 7.5231023E+00 -1.1281727E+04 4.7511987E+03 1.0976043E-01 9.6803904E+00 3.1056233E+03 5.2292749E+03 -2.5029042E-01 1.0654453E+01 -7.5599551E+03 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 1.0600907E+03 -8.7908369E-01 1.8513387E+00 -3.8895288E+03 1.6248770E+03 -1.5172217E-02 2.8376794E+00 -1.4789662E+02 2.2673284E+03 1.8877667E-01 3.9596546E+00 2.5124709E+03 3.9750569E+03 -6.4238346E-01 6.9420252E+00 -1.2322738E+04 5.4398892E+03 -2.8827396E-01 9.5002041E+00 -8.9502500E+03 5.4874829E+03 2.0974435E-01 9.5833206E+00 6.7222744E+03 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 1.1731140E+03 -9.2128611E-01 1.7072686E+00 -4.3906812E+03 1.8484214E+03 -1.0854372E-01 2.6900640E+00 -1.1950244E+03 2.4442273E+03 3.4569389E-01 3.5571585E+00 4.7234243E+03 4.2461313E+03 -5.9229535E-01 6.1795239E+00 -1.2486703E+04 5.6875884E+03 -2.8833213E-01 8.2773209E+00 -9.3594961E+03 6.3592007E+03 -3.2859224E-01 9.2547398E+00 -1.1757339E+04 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 1.2868806E+03 -8.9883047E-01 1.4982692E+00 -4.7671958E+03 2.0593215E+03 -2.2574347E-01 2.3975947E+00 -2.7038494E+03 2.8123025E+03 5.1323569E-01 3.2742634E+00 7.4764253E+03 4.4346304E+03 -4.7134125E-01 5.1630821E+00 -1.1056445E+04 5.8831919E+03 -2.5362745E-01 6.8495908E+00 -8.6101797E+03 6.6293276E+03 -3.7671489E-01 7.7182899E+00 -1.3787740E+04 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 1.4217333E+03 -2.5042098E-02 4.1381840E-02 -2.1353542E+02 2.8758196E+03 -2.2423953E-02 8.3705366E-02 -3.8680191E+02 4.4903926E+03 -1.1577159E-02 1.3070011E-01 -3.1188980E+02 6.0116860E+03 -9.1055688E-03 1.7497981E-01 -3.2842191E+02 8.1402681E+03 1.2691311E-02 2.3693562E-01 -6.1980164E+02 5.4302238E+04 3.4501605E-02 1.5805541E+00 1.1232731E+04 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = K VELOCITY DAMPING FREQUENCY COMPLEX 1.4197896E+03 -3.0325665E-03 4.1325260E-03 -2.5833488E+01 2.8725632E+03 -2.7793243E-03 8.3610583E-03 -4.7902473E+01 4.4856504E+03 -1.7072423E-03 1.3056207E-02 -4.5948467E+01 6.0053628E+03 -1.5231079E-03 1.7479576E-02 -5.4880810E+01 8.1311860E+03 -1.8892016E-03 2.3667125E-02 -9.2168495E+01 6.7762256E+05 3.0369260E-03 1.9723294E+00 1.2347267E+04

EIGENVALUE 6.7137329E+03 1.0874868E+04 2.1162908E+04 2.9409426E+04 3.4713566E+04 5.1588297E+04 EIGENVALUE 7.1930576E+03 1.1690888E+04 2.2286127E+04 3.1959898E+04 3.5929984E+04 5.4428906E+04 EIGENVALUE 7.7129038E+03 1.2657751E+04 2.3378277E+04 3.6177270E+04 3.6356176E+04 5.7928297E+04 EIGENVALUE 8.2733408E+03 1.3829313E+04 2.4362730E+04 3.5927914E+04 4.2539867E+04 5.9925629E+04 EIGENVALUE 8.8795234E+03 1.5283079E+04 2.5249471E+04 3.7165398E+04 4.9029402E+04 6.0985379E+04 EIGENVALUE 9.5482207E+03 1.7131582E+04 2.6067568E+04 3.9184270E+04 5.6759051E+04 6.1341188E+04 EIGENVALUE 1.0315609E+04 1.9496840E+04 2.6853520E+04 4.1982645E+04 6.3359762E+04 6.4797086E+04 EIGENVALUE 1.1245871E+04 2.2083893E+04 2.8120604E+04 4.5584203E+04 6.6244016E+04 7.3444063E+04 EIGENVALUE 1.2435130E+04 2.4256457E+04 3.0941035E+04 4.9389918E+04 6.8971156E+04 7.5710766E+04 EIGENVALUE 1.7056789E+04 3.4503332E+04 5.3882004E+04 7.2137992E+04 9.7677313E+04 6.5133625E+05 EIGENVALUE 1.7037416E+04 3.4470656E+04 5.3827746E+04 7.2064289E+04 9.7574102E+04 8.1314425E+06

Listing 7-10. Output Data for Jet Transport Wing by Strip Theory

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7.5 Flutter Analysis of Jet Transport Wing/Aileron by Strip Theory (Example HA145D) In this example, an aileron is added to the BAH wing of Example HA145C . Strip Theory is used again, but with an approximation to the circulation function, and the PK-method is used. The aileron is assumed to have a stiff actuator and a high aileron frequency (an uncoupled frequency of 60 Hz) so that it does not couple substantially with the wing. The aileron inertial characteristics are given on the CONM1 entry to illustrate inclusion of static unbalance, although the example aileron is balanced (as it should be). The SPLlNE3 entry illustrates the method of including a control surface in Strip Theory. The correction factors [WKK ] shown in Eq. 1-21) are introduced into the theoretical oscillatory aerodynamic coefficients via DMI entries that provide the factors; a 50% reduction in control surface hinge moment is illustrated. The PK-Flutter method is used for comparison to Rodden, Harder, and Bellinger (1979, Figs. 17-19). The aileron characteristics are provided by the input data BAH_AILERON.DAT, shown previously in Listing 6-6. The CAERO4 1001 and PAERO4 entries, along with the AEFACT 1 and AEFACT 2 entries, provide the Strip Theory aerodynamics for nine strips on the wing: five inboard of the aileron, three across the aileron, and one outboard, as shown in Figure 7-8. The PAERO4 entry specifies no Prandtl-Glauert compressibility correction and three terms in the approximate circulation function with coefficients on AEFACT 2. The PAERO4 entry also specifies the control surface geometry on each strip, (there is no control surface on Strips 1 through 5, and 9) and a control surface on Strips 6 through 8 with a 75% hinge line and 25% chord without a gap. The theoretical oscillatory aerodynamic loads are modified by factors on DMI entries. Unit factors are input on each strip for lift and pitching moment and a factor of 0.50 is input to reduce the control surface hinge moments on Strips 6 through 8. Three SPLlNE3 entries provide the control surface connection for the three strips with the control surface. SPLlNE3 151 specifies the relative angle on the sixth strip (UKID = 1006) on panel CAER04 1001 via the code COMP = 6 to be equal to the negative rotation (R2) of GRID 12. SPLINE3 152 and 153 specify the angle on the seventh and eighth strips. The vibration analysis uses the modified Givens method and requests 11 modes: 10 for the wing and one for the aileron. The parameter PARAM,LMODES selects all eleven modes for the modal formulation of the flutter equations. The PK-method is specified on FLUTTER 40 for sea-level density and Mach zero; the tabulation of velocities on FLFACT 4 lists all values as positive so no eigenvectors will be obtained for the six roots that are requested. ENDDATA is the last entry in the input Bulk Data Section.

Case Control Commands The Case Control Section includes the title, subtitle, label, ECHO = BOTH, SPC = 1 for the additional fuselage constraints, SDAMP = 2000 for the structural damping, MPC = 1 for the control surface rotation, METHOD = 10 for the vibration analysis, SVEC = ALL to print the vibration modes of the a-set grid points, and FLUTTER = 40 for the PK-flutter method. SDISP(PLOT) = ALL, and DISP(PLOT) = 10 save the data for subsequent plotting but do not print them, and BEGIN BULK completes the Case Control Section. The Executive Control Section includes ID NXN, HA145D, TIME 5, and SOL 145. CEND concludes the Executive Control Section.

Output The input data are shown in Listing 7-11 and the Sorted Bulk Data entries follow in Listing 7-12. Selected output data are presented in Listing 8-13. Highlights of the output are discussed as follows.

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The statically balanced (and massless) aileron does not affect any of the wing frequencies, and the coupled wing/aileron frequency is 59.993 Hz. The flutter solutions are almost identical to the case without the aileron [Rodden, Harder, and Bellinger (1979, Figs. 17-19)] as expected because of the high aileron frequency, the aileron complete balance, and the reduction in the aileron aerodynamic hinge moment coefficients. In the Flutter Summaries, Points 1 and 2 are of primary interest. Point 1 gives the solution for the stable bending mode until it transitions to an aerodynamic lag root that goes unstable at the divergence speed of V = 1419.9 ft/s. This is the expected divergence speed. The appearance of the aerodynamic lag root was not recognized by Rodden, Harder, and Bellinger (1979) but is explained by Rodden and Bellinger (1982a). Point 2 is the coupled torsion mode that goes unstable at the lowest flutter speed of V = 1133 ft/s (by linear interpolation), as before. The flutter speed is slightly lower than that obtained (1152 ft/s) in the previous Example HA145C of because of the different representations of the circulation function, not because of differences between the K- and PK-methods of flutter analysis.

Figure 7-8. BAH Wing and Aileron and Aerodynamic Strip Idealization N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA145D $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE H145D $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ $ CANTILEVERED WING WITH BALANCED $ $ CONTROL SURFACE $ $ $ $ SOLUTION PK FLUTTER ANALYSIS METHOD USING $ $ STRIP THEORY AERODYNAMICS WITH THE $ $ W.P.JONES APPROXIMATION TO THE $ $ THEODORSEN FUNCTION AT MACH NO. 0. $ $ $ $ OUTPUT TABULATED MODAL DEFLECTIONS PLUS $ $ X-Y PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ TIME 10 $ TIME IN CPU MINUTES SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145D: BAH JET TRANSPORT WING WITH CONTROL SURFACE CANTILEVERED, W.P.JONES STRIP THEORY AT MACH NO. 0.0

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C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145D: BAH JET TRANSPORT WING WITH CONTROL SURFACE 2 SUBTI = CANTILEVERED, W.P.JONES STRIP THEORY AT MACH NO. 0.0 3 LABEL = PK FLUTTER METHOD 4 ECHO = BOTH 5 SPC = 1 $ FUSELAGE CONSTRAINT 6 SDAMP = 2000 $ STRUCTURAL DAMPING 7 MPC = 1 $ CONTROL SURFACE RELATIVE MOTION 8 METHOD = 10 $ MODIFIED GIVENS FOR VIBRATION ANALYSIS 9 SVEC = ALL $ PRINT VIBRATION MODES 10 FMETHOD = 40 $ PK-FLUTTER METHOD 11 $---BE CAREFUL WITH EIGENVECTOR TABULATIONS: THE SDISP COMMAND PRINTS 12 $ ONE MODAL EIGENVECTOR PER PAGE FOR EACH EIGENVALUE AND FOR ALL THE 13 $ VALUES OF REDUCED FREQUENCY; THE DISP COMMAND PRINTS THE CORRESPONDING 14 $ GRID DISPLACEMENTS ONE SET PER PAGE 15 SDISP(PLOT) = ALL $ PLOT AND SAVE ALL MODAL PARTICIPATION FACTORS 16 SET 10 = 1 THRU 5000 17 DISP(PLOT) = 10 $ PLOT AND SAVE STRUCTURAL DISPLACEMENTS IN SET 10 18 OUTPUT(XYOUT) 19 CSCALE 2.0 20 PLOTTER NASTRAN 21 CURVELINESYMBOL = -6 22 YTTITLE = DAMPING G 23 YBTITLE =FREQUENCY F HZ 24 XTITLE = VELOCITY V (FT/SEC) 25 XMIN = 0. 26 XMAX = 2500. 27 YTMIN = -.6 28 YTMAX = +.1 29 YBMIN = 0. 30 YBMAX = 15. 31 XTGRID LINES = YES 32 XBGRID LINES = YES 33 YTGRID LINES = YES 34 YBGRID LINES = YES 35 UPPER TICS = -1 36 TRIGHT TICS = -1 37 BRIGHT TICS = -1 38 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 4(G,F) 5(G,F) 6(G,F) 7(G,F), 39 8(G,F) 9(G,F) 10(G,F) 11(G,F) 12(G,F) 13(G,F) 14(G,F), 40 15(G,F) 16(G,F) 17(G,F) 18(G,F) 19(G,F) 41 CURVELINESYMBOL = -6 42 YTTITLE = DAMPING G 43 YBTITLE =FREQUENCY F HZ 44 XTITLE = VELOCITY V (FT/SEC) 45 XMIN = 0. 46 XMAX = 1600. 47 YTMIN = -1.7 48 YTMAX = +.1 49 YBMIN = 0. 50 YBMAX = 2.5 51 XTGRID LINES = YES 52 XBGRID LINES = YES 53 YTGRID LINES = YES 54 YBGRID LINES = YES 55 UPPER TICS = -1 56 TRIGHT TICS = -1 57 BRIGHT TICS = -1 58 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 4(G,F) 5(G,F) 6(G,F) 7(G,F), 59 8(G,F) 9(G,F) 10(G,F) 11(G,F) 12(G,F) 13(G,F) 14(G,F), 60 15(G,F) 16(G,F) 17(G,F) 18(G,F) 19(G,F) 61 BEGIN BULK EXAMPLE HA145D: BAH JET TRANSPORT WING WITH CONTROL SURFACE PAGE CANTILEVERED, W.P.JONES STRIP THEORY AT MACH NO. 0.0 PK FLUTTER METHOD I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ INCLUDE BAH_STRUCT.DAT $ $ INCLUDE BAH_AILERON.DAT $ $ INCLUDE BAH_MASS.DAT $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $ $ THE SPC ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS $ $ AND ENFORCED DISPLACEMENTS. IT LISTS THE ID, GRID POINT $ $ NO., CONSTRAINED DOFS AND VALUE OF AN ENFORCED DISPLACE$ $ MENT. $

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Flutter Analysis Sample Problems

$

$ $ $ SID G C D SPC 1 11 345 $ $ $ * * STRUCTURAL DAMPING * * $ $ $ $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUSION $ $ OF STRUCTURAL DAMPING IN EQUATIONS OF MOTION (SEE HANDBOOK $ $ FOR DYNAMIC ANALYSIS, SECT. 3.2.2). IF SET TO -1, MODAL $ $ DAMPING IS PUT INTO COMPLEX STIFFNESS MATRIX AS STRUCTURAL $ $ DAMPING. $ $ $ $ N V1 V2 PARAM KDAMP +1 $ $ $ THE TABDMP1 ENTRY DEFINES MODAL DAMPING AS A TABULAR $ $ FUNCTION OF FREQUENCY. THE DAMPING LEVELS ARE LINEAR $ $ BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE EXTRAP$ $ OLATED OUTSIDE THE TABULATED FREQUENCY RANGE. $ $ $ $ ID TABDMP1 2000 +T2000 $ F1 G1 F2 G2 ETC ENDT +T2000 0. 0.0 10. 0.0 ENDT $ $ $ * * * AERODYNAMIC DATA * * * $ $ $ $ (LB-IN-SEC SYSTEM) $ $ $ $ * * ELEMENT GEOMETRY * * $ $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ $ REFERENCE LENGTHS PLUS SYMMETRY KEYS. SYMXZ = 1 INDICATES $ $ THAT THE MODEL IS MOUNTED WITH A ROOT REFLECTION PLANE; $ $ SYMXY = 0 INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH $ $ FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ NOTE THAT THE THE DIMENSIONS OF THE REFERENCE DENSITY OF $ $ AIR ARE SNAILS PER CUBIC INCH, NOT POUNDS PER CUBIC INCH. $ $ $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 131.232 1.1468-7 1 $ THIS CORD2R ENTRY DEFINES THE AERO COORDINATE SYSTEM $ $ FLAGGED BY THE AERO ENTRY. LISTED ARE THE ORIGIN, A $ $ POINT ALONG THE Z AXIS AND A POINT IN THE X-Z PLANE, $ $ ALL IN THE RID COORDINATE SYSTEM. NOTE THAT IN THIS $ $ COORDINATE SYSTEM THE UPSIDE DOWN AND BACKWARDS STRUC$ $ TURE WILL FLY UPSIDE UP AND FORWARD. $ $ $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0. 0. 0. 0. 0. -1. +C1 $ C1 C2 C3 +C1 -1. 0. 0. $ THE CAERO4 ENTRY IS USED TO SPECIFY STRIP THEORY AERO. $ $ PID IDENTIFIES A PAERO4 ENTRY WHERE THE ELEMENT PROPERTIES $ $ ARE GIVEN, CP DEFAULTS TO THE BASIC COORDINATE SYSTEM, $ $ NSPAN WOULD BE THE NUMBER OF EQUAL SPAN STRIPS, HOWEVER $ $ LSPAN SPECIFIES THE AEFACT ENTRY WHERE THE UNEQUAL STRIP $ $ WIDTHS ARE DEFINED. THE CONTINUATION ENTRY DEFINES THE $ $ INBD AND OUTBD LEADING EDGE POINTS AND THE CHORDS. $ $ EID PID CP NSPAN LSPAN CAERO4 1001 1001 0 0 1 +BAH $ X1 Y1 Z1 X12 X4 Y4 Z4 X43 +BAH 78.75 0.0 0.0 225.0 35.0 500.0 0.0 100.0 $ $ $ THIS AEFACT ENTRY GIVES THE EDGES OF THE STRIPS IN PERCENT $ $ OF SPAN. $ $ SID D1 D2 D3 ETC AEFACT 1 .0 .09 .276 .454 .636 .74 .826 +AF +AF .90 .974 1.0 $ $ $ THE PAERO4 ENTRY DEFINES PROPERTIES OF THE STRIP ELEMENTS $ $ FOR STRIP THEORY AERODYNAMICS. IT OFFERS A WIDE VARIETY $ $ OF OPTIONS. CLA DEFAULTS TO NO COMPRESSIBILITY CORRECTION $ $ AND NO CORRECTIONS OF LIFT CURVE SLOPES. LCLA MUST BE ZERO $ $ WHEN CLA IS ZERO. CIRC=2 SPECIFIES AN APPROXIMATE EXPRESSION $ $ FOR THE THEODORSEN CIRCULATION FUNCTION; THE APPOXIMATION $ $ CONSTANTS ARE LISTED ON AN AEFACT ENTRY SPECIFIED BY LCIRC. $ $ DOC, CAOC, AND GAPOC ARE REQUIRED EVEN THOUGH A STRIP HAS NO $ $ CONTROL SURFACE. THESE ARE THE DISTANCE FROM THE QUARTER$ $ CHORD TO THE HINGE LINE, THE CONTROL SURFACE CHORD, AND THE $ $ GAP BETWEEN THE PRIMARY SURFACE AND THE CONTROL SURFACE, ALL $ $ EXPRESSED AS FRACTIONS OF THE LOCAL CHORD. CAOC=0.0 SPECIFIES $ $ NO CONTROL SURFACE. $ $ PID CLA LCLA CIRC LCIRC DOC1 CAOC1 GAPOC1 PAERO4 1001 0 0 2 2 0.0 0.0 0.0 +PA41 +PA41 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 +PA42 +PA42 0.0 0.0 0.0 0.0 0.50 0.25 0.0 0.50 +PA43 +PA43 0.25 0.0 0.50 0.25 0.0 0.0 0.0 0.0 $ THE AEFACT ENTRY IS A UTILITY ENTRY USED TO SPECIFY LISTS OF $ $ NUMBERS. THESE ARE IDENTIFIED BY THE LCIRC FIELD OF THE $ $ PAERO4 ENTRY AND ARE THE W.P.JONES CONSTANTS FOR THE $

NX Nastran Aeroelastic Analysis User’s Guide

7-43

Flutter Analysis Sample Problems

Chapter 7

DMI

7-44

WTFACT

1

$ APPROXIMATE CIRCULATION FUNCTION. $ AEFACT 2 0.0 1.0 -0.165 0.041 -0.335 0.320 $ * AERODYNAMIC CORRECTION FACTORS * $ $ THE DMI ENTRIES PROVIDE FOR DIRECT MATRIX INPUT BY THE USER. $ $ THEY MUST BE ANNOUNCED BY A DMAP ALTER IN THE EXECUTIVE $ $ CONTROL DECK. IN THIS CASE, THEY ARE CORRECTION FACTORS TO $ $ MODIFY THE THEORETICAL OSCILLATORY AERODYNAMIC COEFFICIENTS $ $ TO AGREE WITH EXPERIMENTAL DATA. $ $ NAME "0" FORM TIN TOUT M N DMI WTFACT 0 2 1 21 21 $ 1ST STRIP $ NAME J I1 A(I1,J) A(I1+1,J) ETC. 1 1.0 DMI WTFACT 2 2 1.0 $ 2ND STRIP DMI WTFACT 3 3 1.0 DMI WTFACT 4 4 1.0 $ 3RD STRIP DMI WTFACT 5 5 1.0 DMI WTFACT 6 6 1.0 $ 4TH STRIP DMI WTFACT 7 7 1.0 DMI WTFACT 8 8 1.0 $ 5TH STRIP DMI WTFACT 9 9 1.0 DMI WTFACT 10 10 1.0 $ 6TH STRIP DMI WTFACT 11 11 1.0 DMI WTFACT 12 12 1.0 DMI WTFACT 13 13 0.50 $ 7TH STRIP DMI WTFACT 14 14 1.0 DMI WTFACT 15 15 1.0 DMI WTFACT 16 16 0.50 $ 8TH STRIP DMI WTFACT 17 17 1.0 DMI WTFACT 18 18 1.0 DMI WTFACT 19 19 0.50 $ 9TH STRIP DMI WTFACT 20 20 1.0 DMI WTFACT 21 21 1.0 $ $ $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * BEAM SPLINE FIT ON THE WING * $ $ THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLAT$ $ ION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE $ $ THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS $ $ TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE $ $ DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR $ $ ATTACHMENT AND TORSIONAL FLEXIBILITIES. DTHX AND DTHY $ $ ARE ROTATIONAL ATTACHMENT FLEXIBILITIES. CID IDENTIFIES $ $ THE SPLINE AXIS. $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 100 1001 1002 1009 14 .0 1.0 0 +SP100 $ DTHX DTHY +SP100 -1.0 -1.0 $ $ $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ SID G1 G2 G3 G4 G5 G6 SET1 14 1 THRU 11 $ $ THE SPLINE3 ENTRY DEFINES A CONSTRAINT EQUATION FOR AEROELASTIC $ $ PROBLEMS; IT IS PARTICULARLY USEFUL FOR CONTROL SURFACE $ $ CONSTRAINTS. THE CAERO FIELD IS ID OF MACRO-ELEMENT ON WHICH $ $ THE ELEMENT TO BE INTERPOLATED LIES; UKID FIELD IS ID OF UK $ $ POINT, I.E., THE BOX NUMBER; COMP IS COMPONENT OF MOTION TO BE $ $ INTERPOLATED; GI IS ID OF INDEPENDENT GRID POINT; CI IS $ $ COMPONENT TO BE USED; AI IS COEFFICIENT IN CONSTRAINT $ $ RELATIONSHIP. $ $ $ $ EID CAERO UKID COMP G1 C1 A1 --+ETC SPLINE3 151 1001 1006 6 12 5 -1.0 SPLINE3 152 1001 1007 6 12 5 -1.0 SPLINE3 153 1001 1008 6 12 5 -1.0 $ $ $ * * * SOLUTION SPECIFICATIONS * * * $ $ $ $ * * AERODYNAMIC CONDITIONS * * $ $ $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ $ TO GENERATE GENERALIZED AERO FORCE MATRICES. IF MORE THAN $ $ EIGHT MACH NO.S OR REDUCED FREQUENCIES ARE REQUIRED A SECOND $ $ MKAERO1 ENTRY IS NECESSARY. $ $ $ $ M1 M2 M3 ETC MKAERO1 0.0 +MK1 $ K1 K2 K3 K4 ETC

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

+MK1

0.001

0.02

0.04 0.06 0.08 0.10 0.12 0.20 MKAERO1 0.0 +MK2 0.50 1.0 1.5 $ $ * * SOLUTION PARAMETERS * * $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE (DEFAULT IS 10-3). $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 40 PK 1 2 4 L 6 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NO.S $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 1. FLFACT 2 0.0 FLFACT 4 4800. 6000. 7200. 8400. 9600. 10800. 12000. +FLF4 12597.6 12598.8 13200. 14400. 15600. 17038.8 18000. 19200. +FLF4A 20400. 21600. 22800. 24000. 25200. $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. $ $ N V1 V2 PARAM LMODES 11 PARAM OPGTKG 1 $ $ THE PARAM,VREF,C ENTRY SPECIFIES A CONVERSION FACTOR TO BE $ USED TO CONVERT THE DIMENSIONS OF THE OUTPUT VELOCITIES BY $ DIVIDING BY C, IN THIS CASE BY 12.0 IN/FT TO PRINT VEL$ OCITIES IN FT/SEC RATHER THAN IN/SEC. $ PARAM VREF 12.0 $ $ * * EIGENSOLUTION PARAMETERS * * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM; IN THIS CASE THE $ MODIFIED GIVENS METHOD. ELEVEN MODES ARE DESIRED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 ND EIGR 10 MGIV 11 $ NORM G C +EIGR MAX $ ENDDATA INPUT BULK DATA CARD COUNT = 448

+MK2 $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO +FLF4 +FLF4A RFREQ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +EIGR $ $

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Chapter 7

Flutter Analysis Sample Problems

Listing 7-11. Input Files for Jet Transport Wing/Aileron by Strip Theory EXAMPLE HA145D: BAH JET TRANSPORT WING WITH CONTROL SURFACE APRIL 1, 2004 NX NASTRAN 3/31/93 PAGE 13 CANTILEVERED, W.P.JONES STRIP THEORY AT MACH NO. 0.0 PK FLUTTER METHOD S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 1 .0 .09 .276 .454 .636 .74 .826 +AF 2+AF .90 .974 1.0 3AEFACT 2 0.0 1.0 -0.165 0.041 -0.335 0.320 4AERO 1 131.232 1.1468-71 5CAERO4 1001 1001 0 0 1 +BAH 6+BAH 78.75 0.0 0.0 225.0 35.0 500.0 0.0 100.0 7CELAS2 3 5142661.12 5 8CMASS2 121 5248.7 1 3 9CMASS2 122 134.9 1 3 2 3 10CMASS2 123 790.3 2 3 11CMASS2 341 9727. 3 3 12CMASS2 342 11005. 3 3 4 3 13CMASS2 343 473. 4 3 14CMASS2 561 3253.6 5 3 15CMASS2 562 -139.7 5 3 6 3 16CMASS2 563 946.3 6 3 17CMASS2 781 2617.8 7 3 18CMASS2 782 21. 7 3 8 3 19CMASS2 783 782.3 8 3 20CMASS2 9101 494.8 9 3 21CMASS2 9102 -7.3 9 3 10 3 22CMASS2 9103 185.2 10 3 23CONM1 1 11 +51 24+51 17400. 4.37+7 +52 25+52 4.35+09 26CONM1 2 12 +AIL1 27+AIL1 0.0 0.0 +AIL2 28+AIL2 13970.5 29CORD2R 1 0. 0. 0. 0. 0. -1. +C1 30+C1 -1. 0. 0. 31DMI WTFACT 0 2 1 21 21 32DMI WTFACT 1 1 1.0 33DMI WTFACT 2 2 1.0 34DMI WTFACT 3 3 1.0 35DMI WTFACT 4 4 1.0 36DMI WTFACT 5 5 1.0 37DMI WTFACT 6 6 1.0 38DMI WTFACT 7 7 1.0 39DMI WTFACT 8 8 1.0 40DMI WTFACT 9 9 1.0 41DMI WTFACT 10 10 1.0 42DMI WTFACT 11 11 1.0 43DMI WTFACT 12 12 1.0 44DMI WTFACT 13 13 0.50 45DMI WTFACT 14 14 1.0 46DMI WTFACT 15 15 1.0 47DMI WTFACT 16 16 0.50 48DMI WTFACT 17 17 1.0 49DMI WTFACT 18 18 1.0 50DMI WTFACT 19 19 0.50 51DMI WTFACT 20 20 1.0 52DMI WTFACT 21 21 1.0 53EIGR 10 MGIV 11 +EIGR 54+EIGR MAX 55FLFACT 1 1. DENSITY 56FLFACT 2 0.0 MACH NO 57FLFACT 4 4800. 6000. 7200. 8400. 9600. 10800. 12000. +FLF4 58+FLF4 12597.6 12598.8 13200. 14400. 15600. 17038.8 18000. 19200. +FLF4A 59+FLF4A 20400. 21600. 22800. 24000. 25200. RFREQ 60FLUTTER 40 PK 1 2 4 L 6 61GENEL 432 1 3 2 3 3 3 +01 62+01 4 3 5 3 6 3 7 3 +02 63+02 8 3 9 3 10 3 +03 64+03 UD 11 3 11 4 11 5 +04 65+04 11 6 +05 66+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 67+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 68+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 69+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 70+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 71+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 72+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 73+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 74+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 75+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 76+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 77+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 78+17 413.0 79GRID 1 20.25 90. 12456 80GRID 2 -81. 90. 12456 81GRID 3 17.85 186. 12456 82GRID 4 -71.4 186. 12456 83GRID 5 15.8 268. 12456 84GRID 6 -63.2 268. 12456 NX Nastran Aeroelastic Analysis7User’s Guide 7-46 85GRID 13.3 368. 12456 86GRID 8 -53.2 368. 12456 87GRID 9 11.05 458. 12456

Flutter Analysis Sample Problems

90919293949596979899100101102103104105106107108109110111112113114115-

GRID MKAERO1 +MK2 MKAERO1 +MK1 MPC +MPC1 PAERO4 +PA41 +PA42 +PA43 PARAM PARAM PARAM PARAM PARAM PARAM SET1 SPC SPLINE2 +SP100 SPLINE3 SPLINE3 SPLINE3 TABDMP1 +T2000 ENDDATA TOTAL COUNT=

12 0.0 0.50 0.0 0.001 1

-86.45

368.

1246 +MK2

1.0

1.5 +MK1

1001 0.0 0.0 0.25 GRDPNT KDAMP LMODES OPGTKG VREF WTMASS 14 1 100 -1.0 151 152 153 2000 0.

0.02 0.04 12 3 7 3 0 0 0.0 0.0 0.0 0.0 0.0 0.50 11 +1 11 1 12.0 .0025907 1 THRU 11 345 1001 1002 -1.0 1001 1006 1001 1007 1001 1008

0.06 -1.0 -0.5 2 0.0 0.0 0.25

0.0

0.08 8 12 2 0.0 0.50 0.0

0.10 3 5 0.0 0.0 0.25 0.0

0.12 1.5 33.25 0.0 0.0 0.0 0.0

0.20 0.0 0.0 0.50 0.0

+PA41 +PA42 +PA43

0

+SP100

+MPC1

11 1009

14

.0

1.0

6 6 6

12 12 12

5 5 5

-1.0 -1.0 -1.0

0.0

ENDT

+T2000 10.

116

Listing 7-12. Sorted Bulk Data Entries for Jet Transport Wing/Aileron by Strip Theory EXAMPLE HA145D: BAH JET TRANSPORT WING WITH CONTROL SURFACE PAGE 17 CANTILEVERED, W.P.JONES STRIP THEORY AT MACH NO. 0.0 PK FLUTTER METHOD O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 11 M O * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * 0.000000E+00 0.000000E+00 4.191900E+04 5.128960E+06 -1.642074E+05 0.000000E+00 * 0.000000E+00 0.000000E+00 5.128960E+06 1.350243E+09 -2.381847E+07 0.000000E+00 * 0.000000E+00 0.000000E+00 -1.642074E+05 -2.381847E+07 4.458796E+09 0.000000E+00 * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Y 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Z 4.191900E+04 3.917256E+00 1.223541E+02 0.000000E+00 I(S) * 7.226942E+08 3.727022E+06 0.000000E+00 * * 3.727022E+06 4.458153E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 * I(Q) * 4.458157E+09 * * 7.226906E+08 * * 0.000000E+00 * Q * 9.977400E-04 9.999995E-01 0.000000E+00 * * -9.999995E-01 9.977400E-04 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * R E A L E I G E N V A L U E S EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED ORDER MASS 1 1 1.637768E+02 1.279753E+01 2.036790E+00 8.160930E+00 2 2 4.982469E+02 2.232145E+01 3.552568E+00 5.525822E+01 3 3 2.092550E+03 4.574440E+01 7.280447E+00 7.079897E+00 4 4 5.402874E+03 7.350424E+01 1.169856E+01 8.652712E+00 5 5 8.742090E+03 9.349915E+01 1.488085E+01 4.002357E+00 6 8 1.766007E+04 1.328912E+02 2.115029E+01 3.883447E+00 7 6 2.398459E+04 1.548696E+02 2.464826E+01 3.597392E+00 8 10 4.211864E+04 2.052283E+02 3.266309E+01 3.142601E+00 9 9 6.020812E+04 2.453734E+02 3.905239E+01 1.016253E+00 10 7 9.183204E+04 3.030380E+02 4.823000E+01 8.617019E+00 11 11 1.420885E+05 3.769462E+02 5.999286E+01 3.619337E+01 EIGENVALUE = 1.637768E+02 CYCLES = 2.036790E+00 R E A L E I G E N V E C T O R N O . MODE NO.

POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 12 G EIGENVALUE = 4.982469E+02 CYCLES = 3.552568E+00

T2

R E A L

* * * * * *

GENERALIZED STIFFNESS 1.336571E+03 2.753224E+04 1.481504E+04 4.674951E+04 3.498896E+04 6.858195E+04 8.628196E+04 1.323621E+05 6.118669E+04 7.913184E+05 5.142661E+06 1

T3 R1 R2 R3 4.518474E-02 4.250221E-02 1.482559E-01 1.437271E-01 3.794104E-01 3.767312E-01 6.929946E-01 6.926546E-01 9.989689E-01 1.000000E+00 0.0 User’s Guide NX Nastran Aeroelastic Analysis E I G E N V E C T O R N O . 2

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Chapter 7

Flutter Analysis Sample Problems

POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G -1.635563E-01 2 G 5.449901E-01 3 G -2.796319E-01 4 G 1.000000E+00 5 G -2.267440E-01 6 G 9.465655E-01 7 G -1.388892E-01 8 G 8.878612E-01 9 G -3.973597E-02 10 G 8.224477E-01 12 G 0.0 EIGENVALUE = 1.420885E+05 CYCLES = 5.999286E+01 R E A L E I G E N V E C T O R N O . 11 POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 G 0.0 2 G 0.0 3 G 0.0 4 G 0.0 5 G 0.0 6 G 0.0 7 G 0.0 8 G 0.0 9 G 0.0 10 G 0.0 12 G 1.000000E+00 MATRIX WTFACT (GINO NAME 101 ) IS A DB PREC 21 COLUMN X 21 ROW SQUARE MATRIX. COLUMN 1 ROWS 1 THRU 1 -------------------------------------------------ROW 1) 1.0000D+00 COLUMN 2 ROWS 2 THRU 2 -------------------------------------------------ROW 2) 1.0000D+00 COLUMN 3 ROWS 3 THRU 3 -------------------------------------------------ROW 3) 1.0000D+00 COLUMN 4 ROWS 4 THRU 4 -------------------------------------------------ROW 4) 1.0000D+00 COLUMN 5 ROWS 5 THRU 5 -------------------------------------------------ROW 5) 1.0000D+00 COLUMN 6 ROWS 6 THRU 6 -------------------------------------------------ROW 6) 1.0000D+00 COLUMN 7 ROWS 7 THRU 7 -------------------------------------------------ROW 7) 1.0000D+00 COLUMN 8 ROWS 8 THRU 8 -------------------------------------------------ROW 8) 1.0000D+00 COLUMN 9 ROWS 9 THRU 9 -------------------------------------------------ROW 9) 1.0000D+00 COLUMN 10 ROWS 10 THRU 10 -------------------------------------------------ROW 10) 1.0000D+00 COLUMN 11 ROWS 11 THRU 11 -------------------------------------------------ROW 11) 1.0000D+00 COLUMN 12 ROWS 12 THRU 12 -------------------------------------------------ROW 12) 1.0000D+00 COLUMN 13 ROWS 13 THRU 13 -------------------------------------------------ROW 13) 5.0000D-01 COLUMN 14 ROWS 14 THRU 14 -------------------------------------------------ROW 14) 1.0000D+00 COLUMN 15 ROWS 15 THRU 15 -------------------------------------------------ROW 15) 1.0000D+00 COLUMN 16 ROWS 16 THRU 16 -------------------------------------------------ROW 16) 5.0000D-01 COLUMN 17 ROWS 17 THRU 17 -------------------------------------------------ROW 17) 1.0000D+00 COLUMN 18 ROWS 18 THRU 18 -------------------------------------------------ROW 18) 1.0000D+00 COLUMN 19 ROWS 19 THRU 19 -------------------------------------------------ROW 19) 5.0000D-01 COLUMN 20 ROWS 20 THRU 20 -------------------------------------------------ROW 20) 1.0000D+00 COLUMN 21 ROWS 21 THRU 21 -------------------------------------------------ROW 21) 1.0000D+00 THE NUMBER OF NON-ZERO TERMS IN THE DENSEST COLUMN = 1 THE DENSITY OF THIS MATRIX IS 4.76 PERCENT.

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Flutter Analysis Sample Problems

FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.1766 5.6623220E+00 4.0000000E+02 -2.4420439E-01 2.0561607E+00 -1.5774674E+00 1.2919239E+01 0.1415 7.0673518E+00 5.0000000E+02 -3.2955328E-01 2.0592303E+00 -2.1319668E+00 1.2938526E+01 0.1178 8.4884424E+00 6.0000000E+02 -4.3011853E-01 2.0573816E+00 -2.7800519E+00 1.2926910E+01 0.1004 9.9567423E+00 7.0000000E+02 -5.5466795E-01 2.0463145E+00 -3.5657868E+00 1.2857374E+01 0.0865 1.1554445E+01 8.0000000E+02 -7.1936250E-01 2.0152667E+00 -4.5543900E+00 1.2662294E+01 0.0742 1.3476210E+01 9.0000000E+02 -9.5377046E-01 1.9438661E+00 -5.8245196E+00 1.2213672E+01 0.0611 1.6379684E+01 1.0000000E+03 -1.3574114E+00 1.7769945E+00 -7.5778751E+00 1.1165186E+01 0.0528 1.8946871E+01 1.0497999E+03 -1.7367283E+00 1.6127264E+00 -8.7991858E+00 1.0133060E+01 0.0000 9.9999996E+24 1.0499000E+03 -1.9979893E-01 0.0000000E+00 -1.3295465E+01 0.0000000E+00 0.0000 9.9999996E+24 1.1000000E+03 -1.0474118E-01 0.0000000E+00 -7.3025169E+00 0.0000000E+00 0.0000 9.9999996E+24 1.2000000E+03 -5.2409641E-02 0.0000000E+00 -3.9861615E+00 0.0000000E+00 0.0000 9.9999996E+24 1.3000000E+03 -2.3549685E-02 0.0000000E+00 -1.9403983E+00 0.0000000E+00 0.0000 9.9999996E+24 1.4199000E+03 -1.6661106E-04 0.0000000E+00 -1.4994225E-02 0.0000000E+00 0.0000 9.9999996E+24 1.5000000E+03 1.1860256E-02 0.0000000E+00 1.1275811E+00 0.0000000E+00 0.0000 9.9999996E+24 1.6000000E+03 2.4461586E-02 0.0000000E+00 2.4806590E+00 0.0000000E+00 0.0000 9.9999996E+24 1.7000000E+03 3.5293423E-02 0.0000000E+00 3.8028150E+00 0.0000000E+00 0.0000 9.9999996E+24 1.8000000E+03 4.4955902E-02 0.0000000E+00 5.1288700E+00 0.0000000E+00 0.0000 9.9999996E+24 1.9000000E+03 5.3828783E-02 0.0000000E+00 6.4823227E+00 0.0000000E+00 0.0000 9.9999996E+24 2.0000000E+03 6.2168136E-02 0.0000000E+00 7.8806186E+00 0.0000000E+00 0.0000 9.9999996E+24 2.1000000E+03 7.0166565E-02 0.0000000E+00 9.3392496E+00 0.0000000E+00 POINT = 2 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.2987 3.3474324E+00 4.0000000E+02 -3.9937280E-02 3.4780819E+00 -4.3638334E-01 2.1853434E+01 0.2360 4.2375751E+00 5.0000000E+02 -5.0273400E-02 3.4343474E+00 -5.4241586E-01 2.1578642E+01 0.1934 5.1695476E+00 6.0000000E+02 -5.8159672E-02 3.3782387E+00 -6.1725152E-01 2.1226101E+01 0.1625 6.1536241E+00 7.0000000E+02 -6.1284974E-02 3.3109965E+00 -6.3747424E-01 2.0803606E+01 0.1390 7.1942186E+00 8.0000000E+02 -5.7188056E-02 3.2366669E+00 -5.8150470E-01 2.0336578E+01 0.1207 8.2822332E+00 9.0000000E+02 -4.4563793E-02 3.1629088E+00 -4.4281128E-01 1.9873142E+01 0.1064 9.3984213E+00 1.0000000E+03 -2.5869235E-02 3.0969679E+00 -2.5169244E-01 1.9458824E+01 0.1004 9.9602737E+00 1.0497999E+03 -1.5862148E-02 3.0677991E+00 -1.5287583E-01 1.9275551E+01 0.1004 9.9615774E+00 1.0499000E+03 -1.5506835E-02 3.0676894E+00 -1.4944606E-01 1.9274862E+01 0.0950 1.0529980E+01 1.1000000E+03 -5.5110450E-03 3.0405822E+00 -5.2642994E-02 1.9104542E+01 0.0857 1.1674541E+01 1.2000000E+03 1.2074869E-02 2.9918036E+00 1.1349204E-01 1.8798058E+01 0.0779 1.2835329E+01 1.3000000E+03 2.4647040E-02 2.9480033E+00 2.2826675E-01 1.8522852E+01 0.0702 1.4254667E+01 1.4199000E+03 2.8739441E-02 2.8992944E+00 2.6177040E-01 1.8216805E+01 0.0657 1.5218550E+01 1.5000000E+03 2.6729194E-02 2.8688617E+00 2.4090476E-01 1.8025591E+01 0.0608 1.6439577E+01 1.6000000E+03 1.5277724E-02 2.8328331E+00 1.3596575E-01 1.7799215E+01 0.0566 1.7676716E+01 1.7000000E+03 -8.5209534E-03 2.7992325E+00 -7.4933678E-02 1.7588097E+01 0.0528 1.8925205E+01 1.8000000E+03 -5.0853573E-02 2.7683661E+00 -4.4227752E-01 1.7394157E+01 0.0495 2.0184956E+01 1.9000000E+03 -1.1562889E-01 2.7397909E+00 -9.9525338E-01 1.7214615E+01 0.0465 2.1496552E+01 2.0000000E+03 -2.1412642E-01 2.7080259E+00 -1.8216838E+00 1.7015030E+01 0.0434 2.3034266E+01 2.1000000E+03 -3.6526951E-01 2.6536067E+00 -3.0450881E+00 1.6673103E+01 POINT = 3 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.6214 1.6091905E+00 4.0000000E+02 -4.9142484E-02 7.2350936E+00 -1.1169947E+00 4.5459435E+01 0.4983 2.0068948E+00 5.0000000E+02 -6.5004468E-02 7.2516527E+00 -1.4809148E+00 4.5563480E+01 0.4162 2.4028482E+00 6.0000000E+02 -8.2334675E-02 7.2680273E+00 -1.8799626E+00 4.5666363E+01 0.3575 2.7971790E+00 7.0000000E+02 -1.0112985E-01 7.2839913E+00 -2.3141882E+00 4.5766670E+01 0.3135 3.1901231E+00 8.0000000E+02 -1.2165799E-01 7.2991819E+00 -2.7897463E+00 4.5862114E+01 0.2792 3.5823009E+00 9.0000000E+02 -1.4434153E-01 7.3126035E+00 -3.3159900E+00 4.5946445E+01 0.2516 3.9747384E+00 1.0000000E+03 -1.6996393E-01 7.3228993E+00 -3.9101164E+00 4.6011135E+01 0.2398 4.1707745E+00 1.0497999E+03 -1.8402000E-01 7.3262458E+00 -4.2354193E+00 4.6032162E+01 0.2397 4.1711688E+00 1.0499000E+03 -1.8404870E-01 7.3262506E+00 -4.2360826E+00 4.6032192E+01 0.2289 4.3693166E+00 1.1000000E+03 -1.9958718E-01 7.3277521E+00 -4.5946593E+00 4.6041626E+01 0.2096 4.7704153E+00 1.2000000E+03 -2.3574747E-01 7.3217797E+00 -5.4226751E+00 4.6004101E+01 0.1927 5.1888719E+00 1.3000000E+03 -2.8228080E-01 7.2922583E+00 -6.4668570E+00 4.5818611E+01 0.1735 5.7622743E+00 1.4199000E+03 -3.6187363E-01 7.1722507E+00 -8.1538420E+00 4.5064583E+01 0.1593 6.2764869E+00 1.5000000E+03 -4.3319976E-01 6.9561071E+00 -9.4668255E+00 4.3706512E+01 0.1390 7.1964941E+00 1.6000000E+03 -4.9804926E-01 6.4712863E+00 -1.0125415E+01 4.0660294E+01 0.1214 8.2375851E+00 1.7000000E+03 -4.7451165E-01 6.0067649E+00 -8.9544191E+00 3.7741619E+01 0.1088 9.1920557E+00 1.8000000E+03 -4.0002409E-01 5.6996932E+00 -7.1628771E+00 3.5812229E+01 0.0989 1.0114189E+01 1.9000000E+03 -3.0531022E-01 5.4678192E+00 -5.2445159E+00 3.4355324E+01 0.0903 1.1070879E+01 2.0000000E+03 -1.9264352E-01 5.2582293E+00 -3.1823196E+00 3.3038429E+01 0.0826 1.2109256E+01 2.1000000E+03 -5.5049565E-02 5.0476995E+00 -8.7296593E-01 3.1715633E+01 POINT = 4 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.9954 1.0046399E+00 4.0000000E+02 -2.5909605E-02 1.1588873E+01 -9.4330442E-01 7.2815041E+01 0.7976 1.2537988E+00 5.0000000E+02 -3.3145003E-02 1.1607368E+01 -1.2086531E+00 7.2931244E+01 0.6658 1.5019433E+00 6.0000000E+02 -4.1014399E-02 1.1627580E+01 -1.4982200E+00 7.3058243E+01 0.5718 1.7488163E+00 7.0000000E+02 -4.9539886E-02 1.1650524E+01 -1.8132195E+00 7.3202408E+01 0.5015 1.9939029E+00 8.0000000E+02 -5.8691785E-02 1.1678246E+01 -2.1533015E+00 7.3376587E+01 0.4471 2.2363906E+00 9.0000000E+02 -6.8367414E-02 1.1713494E+01 -2.5158546E+00 7.3598061E+01 0.4041 2.4745743E+00 1.0000000E+03 -7.8291900E-02 1.1762270E+01 -2.8930628E+00 7.3904526E+01 0.3860 2.5903931E+00 1.0497999E+03 -8.2976714E-02 1.1795939E+01 -3.0749540E+00 7.4116074E+01 0.3860 2.5906236E+00 1.0499000E+03 -8.2984276E-02 1.1796012E+01 -3.0752535E+00 7.4116531E+01 0.3698 2.7040808E+00 1.1000000E+03 -8.6826794E-02 1.1840352E+01 -3.2297451E+00 7.4395126E+01 0.3424 2.9201806E+00 1.2000000E+03 -8.3763912E-02 1.1960880E+01 -3.1475306E+00 7.5152428E+01 0.3110 3.2153144E+00 1.3000000E+03 -2.8281605E-01 1.1768241E+01 -1.0455997E+01 7.3942039E+01 0.2686 3.7228286E+00 1.4199000E+03 -3.1635937E-01 1.1101364E+01 -1.1033339E+01 6.9751930E+01 0.2436 4.1048670E+00 1.5000000E+03 -3.0202147E-01 1.0636134E+01 -1.0091866E+01 6.6828804E+01 0.2186 4.5751591E+00 1.6000000E+03 -2.2783104E-01 1.0179007E+01 -7.2856479E+00 6.3956589E+01 0.2001 4.9973831E+00 1.7000000E+03 -1.3295659E-01 9.9014301E+00 -4.1357822E+00 6.2212521E+01 0.1846 5.4175787E+00 1.8000000E+03 -5.5197932E-02 9.6707220E+00 -1.6769943E+00 6.0762939E+01 0.1707 5.8579936E+00 1.9000000E+03 1.5998115E-03 9.4405289E+00 4.7447685E-02 5.9316597E+01 0.1581 6.3244147E+00 2.0000000E+03 3.9703224E-02 9.2045231E+00 1.1480926E+00 5.7833725E+01 0.1467 6.8187490E+00 2.1000000E+03 6.1259042E-02 8.9640903E+00 1.7251478E+00 5.6323044E+01

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Listing 7-13. Output for Jet Transport Wing/Aileron by Strip Theory

7.6 Subsonic Flutter Analysis of the 15-Degree Sweptback Wing by the KE-Method (Example HA145E) The subsonic flutter characteristics of several flat-plate flutter models were reported by Tuovila and McCarty (1955) and investigated further by Yates and Bennett (1963). The untapered wing with 15 deg of sweepback was analyzed by Rodden, Harder, and Bellinger (1979) with its structure idealized as a “stick” model and with aerodynamic forces calculated by the Doublet-Lattice method. Both the calculated flutter speed and frequency were higher than test results in the earlier three-mode solution. The purpose of this example is to reconsider the analysis of Rodden, Harder, and Bellinger (1979) with a structural plate model and to demonstrate the KE-method of flutter analysis. Representation of the model by CQUAD4 elements is more realistic. Such a structural model was developed for the static aeroelastic analysis in Example HA144C . For the static analysis, nominal values of moduli of elasticity and density of aluminum were used and a rigidly clamped support was assumed. However, modifications are appropriate in view of the experimental data. The nominal density of 0.100 lb/in3 results in a model weight of 0.041041 lb, whereas the measured weight was 0.040 lb. Therefore, an adjusted density of 0.097464 lb/in3 is used. The other experimental data are the first three vibration frequencies and node lines. The structural root model of Example HA144C was modified to investigate the effect of constraints on the second (torsion) node line. The RBAR was removed and Grid Points 9, 17, and 25 were given single-point constraints, and variations in root constraints were investigated. A free GRID 17 resulted in an improved correlation with the node lines of Yates and Bennett (1963), therefore GRIDs 9 and 25 are assumed to be fully constrained and GRID 17 is assumed unconstrained in the flutter analysis. With the root so constrained and the adjusted density, the nominal values of moduli, E = 10.3 × 103psi and G = 3.9 × 106psi, resulted in a second vibration frequency of 221.697 Hz as compared to the experimental value of 210 Hz. The moduli were then scaled by the square of the frequency ratio to be E = 9.2418 × 106psi and G = 3.4993 × 106psi. It is not likely that the material moduli were so low; it is more likely that there was additional flexibility in the model support system. The adjusted moduli result in the first three natural frequencies of 34.3, 210.0, and 260.4 Hz that may be compared to the experimental values of 36.0, 210.0, and 242.0 Hz. The variation in experimental frequencies is indicated by the measured frequencies on the similar aluminum model for the Mach 1.3 test that were 36.0, 210.0, and 254.0 Hz.

Bulk Data Entries The Bulk Data Section builds on that of Example HA144C for the basic structural and aerodynamic data. The parameter PARAM,GRDPNT,17 has been added to obtain the weight data that permitted adjusting the density. The KE-method of flutter analysis is being demonstrated in this example. Since the KE-method permits inclusion of damping only as a complex frequency dependent factor in the structural stiffness, structural damping is included by means of the parameter PARAM,KDAMP,-1 and the TABDMP1 table. No damping data were reported by Tuovila and McCarty (1955) so a value of g = 0.01 is assumed up to a frequency of 1000.0 Hz. The aerodynamic data begin with the AERO entry that uses the basic coordinate system for the aerodynamic coordinate system, a reference chord of 2.0706 in., sea-level reference density, SYMXZ = 1 for a complete wind tunnel wall reflection, and SYMXY = 0 to neglect wind tunnel floor or ceiling interference. The CAERO1, PAERO1, SPLlNE1 and SET1 entries provide the aerodynamic geometry and surface spline data and are the same as in Example HA144C . The

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MKAERO1 entry provides the data required to interpolate the aerodynamic matrices in the flutter solution. The Mach number, m = 0.45, and reduced frequencies, k = 0.001, 0.10, 0.12, 0.14, 0.16 and 0.20, are specified to encompass the experimental value of k = 0.1314. The Modified Givens method is selected for vibration analysis on the EIGR entry to obtain four modes that have their largest components normalized to unity. The parameter PARAM,LMODES,4 specifies that all four vibration modes be included in the formulation of the flutter problem. The FLUTTER entry specifies the KE-method of flutter analysis. Note that no EIGC entry is required since the Upper Hessenberg method is built into the KE-method for complex eigenvalues. The FLUTTER entry also identifies the three FLFACT entries that list the densities, Mach numbers and reduced frequencies, and specifies linear interpolation for the aerodynamic generalized force influence coefficients. The data item NVALUE is left blank so the KE-method computes the default LMODES complex eigenvalues. FLFACT 1 gives the density ratio for the wind tunnel test, FLFACT 2 gives the test Mach number, and FLFACT 3 gives the series of reduced frequencies for the flutter analysis. The output flutter velocities in the summary are converted from the analysis units of in/s to ft/s by means of the parameter PARAM,VREF,12.0. The Bulk Data Section ends with the ENDDATA entry.

Case Control Commands The first three entries in the Case Control Section are titles. ECHO = BOTH requests the input data be printed in unsorted and sorted formats. SPC = 1 selects the constraints on the wing root deflections and on the plate element in-plane rotations. SDAMP = 2000 invokes the tabular Bulk Data for structural damping. METHOD = 10 selects the eigenvalue method for the vibration analysis, and FMETHOD = 30 selects the method of flutter analysis. SVEC = ALL requests printed output of the displacements of the grid point degrees of freedom in the analysis set (a-set), which in this case are the transverse deflections in the vibration modes. The OUTPUT(PLOT) entry identifies the plot packet for the structure plotter. CSCALE 2.0 specifies spacing of characters in the plot title. The set of elements to be plotted are the QUAD4s. The FIND entry determines a scale and origin for the plots. The PLOT entry requests plots of the vibration modes with undeformed underlays. The OUTPUT(XYOUT) entry identifies the plot packet for xy-plots. CURVELINESYMBOL = 6 requests points on the first curve to be identified by circle symbols. XYPLOT VG/1 (G,F) 2(G,F) 3(G,F) requests two types of xy-plots: plots of velocity (x-axis) versus damping (y-axis) on the upper half of each frame and velocity versus frequency (y-axis) on the lower half of each frame. Three curves are requested and represent the first three modes. YTTITLE and YBTITLE request titles for the top and bottom plots. Similarly, XTGRID, etc. request that reference lines be drawn on the plots at equal increments. BEGIN BULK completes the Case Control Section. The Executive Control Section begins with the problem identification, ID NXN, HA145E. TIME 5 limits the CPU time to 5.0 minutes, and SOL 145 requests the Structured Aerodynamic Flutter DMAP sequence. The CEND entry completes the Executive Control Section.

Output The input data for this example are shown in Listing 7-14 and Listing 7-15 and limited output data follow in Listing 7-16. The output data are discussed below. The output consists of the weight and inertial data (which were used to monitor the model weight and adjust the structural density), the vibration frequencies and the four (LMODES = 4) vibration modal deflections for the a-set grid points, and the Flutter Summaries. The sorting algorithm in the KE-method is shown in Eq. 1-126 and Eq. 1-127 and the “Points” in the Summary Tables are not necessarily in the order of ascending modes. In this example, Point 1 is

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the third mode, Point 2 is the second (critical) mode, Point 3 is the fourth mode, and Point 4 is the first mode. This is the order in which the complex eigenvalues for the first value of reduced frequency were output from the eigenvalue subroutine. The flutter data from Point 2 are plotted in Figure 7-9 along with the experimental data. The present four mode solution gives a flutter speed of Vf = 483 ft/sec and flutter frequency of ff = 113.0 Hz. The test results were Vf = 495 ft/sec and ff = 120 Hz. The earlier analysis of Rodden, Harder, and Bellinger (1979, Fig. 14), using the structural stick model, and three vibration modes obtained Vf = 509 ft/sec and ff = 134 Hz without any adjustment of that model to agree with the measured vibration frequencies or including any structural damping.

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Figure 7-9. Critical Subsonic Flutter V-g and V-f Curvesfor 15-Degree Sweptback Wing NX Nastran Aeroelastic Analysis User’s Guide

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N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA145E $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145E $$$$$$$$ $ $ $ MODEL DESCRIPTION MODEL A OF NASA TN D-1824 $ $ HALF SPAN 15 DEGREE SWEPT WING $ $ 28 QUAD4 PANEL MODEL $ $ $ $ SOLUTION KE FLUTTER ANALYSIS METHOD $ $ USING DOUBLET LATTICE METHOD $ $ AERODYNAMICS AT MACH NO. 0.45 $ $ $ $ OUTPUT PLOTS OF VIBRATION MODES AND X-Y $ $ PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145E: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 2 KE-METHOD FLUTTER ANALYSIS, DOUBLET-LATTICE AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145E: HALF SPAN 15-DEG SWEPT UNTAPERED WING 2 SUBT = KE-METHOD FLUTTER ANALYSIS, DOUBLET-LATTICE AERO 3 LABEL = 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES 4 ECHO = BOTH 5 SPC = 1 $ WING ROOT DEFLECTIONS AND PLATE IN-PLANE ROTATIONS FIXED 6 SDAMP = 2000 7 METHOD = 10 $ MODIFIED GIVENS METHOD OF REAL EIGENVALUE EXTRACTION 8 FMETHOD = 30 $ KE-FLUTTER METHOD 9 SVEC = ALL $ PRINT VIBRATION MODES 10 OUTPUT(PLOT) 11 CSCALE 2.0 12 PLOTTER NASTRAN 13 SET 1 = QUAD4 14 PTITLE = STRUCTURAL ELEMENTS 15 FIND SCALE, ORIGIN 1, SET 1 16 PLOT MODAL 0 ORIGIN 1, SET 1 17 OUTPUT (XYOUT) 18 CSCALE 2.0 19 PLOTTER NASTRAN 20 CURVELINESYMBOL = 6 21 YTTITLE = DAMPING G 22 YBTITLE = FREQUENCY F HZ 23 XTITLE = VELOCITY V (FT/S) 24 XTGRID LINES = YES 25 XBGRID LINES = YES 26 YTGRID LINES = YES 27 YBGRID LINES = YES 28 UPPER TICS = -1 29 TRIGHT TICS = -1 30 BRIGHT TICS = -1 31 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 32 BEGIN BULK EXAMPLE HA145E: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 3 KE-METHOD FLUTTER ANALYSIS, DOUBLET-LATTICE AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ INCLUDE PLATE_STRUCT.DAT $ $ * * MASS AND INERTIA PROPERTIES * * $ $ $ $ THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES FOR LINEAR, $ $ TEMPERATURE-INDEPENDENT, ISOTROPIC MATERIALS. IT LISTS $ $ YOUNGS MODULUS, THE SHEAR MODULUS, POISSONS RATIO, MASS $ $ DENSITY, THERMAL EXPANSION COEFFICIENT AND STRUCTURAL $ $ ELEMENT DAMPING. $ $ $ $ MID E G NU RHO A TREF GE MAT1 1 9.2418+63.4993+6 0.097464 $ $ PARAM GRDPNT CAUSES THE INERTIAL MATRIX TO BE GENERATED WITH $ $ RESPECT TO COORDINATES THROUGH THE GRID POINT SPECIFIED. $ $ PARAM GRDPNT 17 $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $

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Flutter Analysis Sample Problems

$

$ THE SPC1 ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS. $ IT LISTS THE DOF COMPONENTS TO BE CONSTRAINED AND THE GRID $ POINT NO. $ $ SID C G1 G2 ETC SPC1 1 12345 9 SPC1 1 12345 25 SPC1 1 6 1 THRU 40 $ $ THE ASET1 ENTRY DEFINES DOFS THAT THE USER DESIRES TO BE $ IN THE ANALYSIS SET. IT LISTS THE DOF COMPONENT NUMBERS $ AND THE GRID ID NOS. $ $ V ID1 THRU ID2 ASET1 3 1 THRU 8 ASET1 3 10 THRU 16 ASET1 3 18 THRU 24 ASET1 3 26 THRU 40 $ $ * STRUCTURAL DAMPING * $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUDING $ STRUCTURAL DAMPING IN THE EQUATIONS OF MOTION (SEE THE $ HANDBOOK FOR DYNAMIC ANALYSIS, SECTION 3.2.2). IF SET TO $ -1, MODAL STRUCTURAL DAMPING IS INCLUDED AS THE IMAGINARY $ PART OF A COMPLEX STIFFNESS MATRIX. $ N V1 V2 PARAM KDAMP -1 $ $ THE TABDMP1 ENTRY AND TABLE DEFINE MODAL DAMPING AS A $ TABULAR FUNCTION OF FREQUENCY. THE DAMPING VALUES ARE $ LINEAR BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE $ EXTRAPOLATED OUTSIDE THE TABULATED FREQUENCY RANGE. $ ID TABDMP1 2000 $ F1 G1 F2 G2 ETC ENDT +T2000 0.0 0.01 1000.0 0.01 ENDT $ * * * AERODYNAMIC DATA * * * $ $ $ (LB-IN-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD $ AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=0 INDICATES $ THAT THE MODEL IS MOUNTED AS AN ISOLATED WING; SYMXY = 0 $ ASSUMES THAT THE MODEL IS MOUNTED FAR ENOUGH FROM THE $ FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 0 2.0706 1.1092-7 1 $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AERODYNAMICS. $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS. $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ FOR NON-UNIFORM SPACING. IGID IS THE ID OF ITS ASSOCIATED $ INTERFERENCE GROUP. THE CONTINUATION ENTRY DEFINES POINTS $ 1 AND 4, THE ROOT CHORD AND TIP CHORD. THE BOXES FORMED $ BY THE GRID LINES WILL BE NUMBERED BEGINNING WITH EID $ SO CHOOSE A NUMBER THAT IS UNIQUE, AND IS GREATER THAN ALL $ STRUCTURAL GRID, SCALAR AND EXTRA POINT IDS. $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 101 1 0 6 4 1 $ ( FWD LEFT POINT ) ROOTCHORD ( FWD RIGHT POINT ) TIP CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1 $ $ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ * SURFACE SPLINE FIT ON THE WING * $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $ GRID POINTS ON THE SETG ENTRY TO THE SUB-REGION DEFINED $ BY AERODYNAMIC BOXES 101 THRU 124 OF THE REGION ON THE $ CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE $ SPLINE IS TO BE IMPOSED. $ $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 100 101 101 124 100 .0 $ THE SET1 ENTRY DEFINES THE SET OF POINTS TO BE USED BY

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $

$ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ +TDP +T2000 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CA101

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

7-55

Chapter 7

Flutter Analysis Sample Problems

$ $ $ SET1 +S1 +S2 $ $ $ $ $

$

7-56

THE SURFACE SPLINE FOR INTERPOLATION. SID 100 15 31

G1 2 18 34

G2 4 20 36

G3 6 22 38

G4 8 24 40

$ $ ETC 9 25

11 27

13 29

* * AERODYNAMIC DATABASE * * ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED TO GENERATE GENERALIZED AERO FORCE MATRICES. $ M1 M2 M3 ETC MKAERO1 0.45 $ K1 K2 K3 ETC +MK .001 0.10 0.12 0.14 0.16 0.20 $ * * VIBRATION ANALYSIS * * $ $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE; IN THIS CASE THE MODIFIED $ GIVENS METHOD. SIX MODES ARE DESIRED, NORMALIZED ON THE $ MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 NO EIGR 10 MGIV 4 $ NORM +ER MAX $ $ THE PARAM,OPPHIPA,1 PROVIDES THE VIBRATION MODES FOR THE $ MODAL PLOT REQUESTS. $ PARAM OPPHIPA 1 $ $ * * FLUTTER ANALYSIS * * $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N VIBRATION MODES $ ARE TO BE USED IN THE FLUTTER ANALYSIS. IF A RESTART IS $ CONTEMPLATED TO DELETE MODES FROM THE ANALYSIS LMODES $ SHOULD BE LARGE ENOUGH TO ACCOMMODATE THE DELETION. $ PARAM LMODES 4 $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE. $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 30 KE 1 2 3 L $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS. $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 0.967 FLFACT 2 .45 FLFACT 3 .20000 .16667 .14286 .12500 .11111 .10000 $ $ THE PARAMETER VREF IS USED TO DIVIDE THE ANALYSIS UNITS $ OF VELOCITY TO THOSE DESIRED IN THE FLUTTER SUMMARY, IN $ THIS CASE FROM IN/S TO FT/S. $ $ N V1 V2 PARAM VREF 12.0 $ ENDDATA INPUT BULK DATA CARD COUNT = 321

NX Nastran Aeroelastic Analysis User’s Guide

+S1 +S2 $ $ $ $ $ +MK $ $ $ $ $ $ $ $ +ER $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO RFREQ $ $ $ $ $

Flutter Analysis Sample Problems

Listing 7-14. Input Data for the 15-Degree Sweptback Wing by KE-Method EXAMPLE HA145E: HALF SPAN 15-DEG SWEPT UNTAPERED WING KE-METHOD FLUTTER ANALYSIS, DOUBLET-LATTICE AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES S O R T E D CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 1AERO 0 2.0706 1.1092-71 2ASET1 3 1 THRU 8 3ASET1 3 10 THRU 16 4ASET1 3 18 THRU 24 5ASET1 3 26 THRU 40 6CAERO1 101 1 0 6 4 7+CA101 .0 .0 .0 2.07055 1.48044 8CQUAD4 1 1 1 2 10 9+M00000 0.0 0.0 .041 10CQUAD4 2 1 2 3 11 11+M00001 0.0 0.0 .041 12CQUAD4 3 1 3 4 12 13+M00002 0.0 0.0 .041 14CQUAD4 4 1 4 5 13 15+M00003 0.0 0.0 .041 16CQUAD4 5 1 5 6 14 17+M00004 0.0 0.0 .041 18CQUAD4 6 1 6 7 15 19+M00005 0.0 0.0 .041 20CQUAD4 7 1 7 8 16 21+M00006 0.0 0.0 .041 22CQUAD4 8 1 9 10 18 23CQUAD4 9 1 10 11 19 24CQUAD4 10 1 11 12 20 25CQUAD4 11 1 12 13 21 26CQUAD4 12 1 13 14 22 27CQUAD4 13 1 14 15 23 28CQUAD4 14 1 15 16 24 29CQUAD4 15 1 17 18 26 30CQUAD4 16 1 18 19 27 31CQUAD4 17 1 19 20 28 32CQUAD4 18 1 20 21 29 33CQUAD4 19 1 21 22 30 34CQUAD4 20 1 22 23 31 35CQUAD4 21 1 23 24 32 36CQUAD4 22 1 25 26 34 37+M00007 .041 .041 0.0 38CQUAD4 23 1 26 27 35 39+M00008 .041 .041 0.0 40CQUAD4 24 1 27 28 36 41+M00009 .041 .041 0.0 42CQUAD4 25 1 28 29 37 43+M00010 .041 .041 0.0 44CQUAD4 26 1 29 30 38 45+M00011 .041 .041 0.0 46CQUAD4 27 1 30 31 39 47+M00012 .041 .041 0.0 48CQUAD4 28 1 31 32 40 49+M00013 .041 .041 0.0 50EIGR 10 MGIV 51+ER MAX 52FLFACT 1 0.967 53FLFACT 2 .45 54FLFACT 3 .20000 .16667 .14286 .12500 55FLUTTER 30 KE 1 2 3 56GRID 1 0.0 0.0 0.0 57GRID 2 .211491 .7893 0.0 58GRID 3 .422983 1.5786 0.0 59GRID 4 .634474 2.3679 0.0 60GRID 5 .845966 3.1572 0.0 61GRID 6 1.05746 3.9465 0.0 62GRID 7 1.26895 4.7358 0.0 63GRID 8 1.48044 5.5251 0.0 64GRID 9 .258819 0.0 0.0 65GRID 10 .47031 .7893 0.0 66GRID 11 .681802 1.5786 0.0 67GRID 12 .893293 2.3679 0.0 68GRID 13 1.10478 3.1572 0.0 69GRID 14 1.31628 3.9465 0.0 70GRID 15 1.52777 4.7358 0.0 71GRID 16 1.73926 5.5251 0.0 72GRID 17 1.03528 0.0 0.0 73GRID 18 1.24677 .7893 0.0 74GRID 19 1.45826 1.5786 0.0 75GRID 20 1.66975 2.3679 0.0 76GRID 21 1.88124 3.1572 0.0 77GRID 22 2.09273 3.9465 0.0 78GRID 23 2.30422 4.7358 0.0 79GRID 24 2.51572 5.5251 0.0 80GRID 25 1.81173 0.0 0.0 81GRID 26 2.02322 .7893 0.0 82GRID 27 2.23471 1.5786 0.0 83GRID 28 2.44621 2.3679 0.0 84GRID 29 2.6577 3.1572 0.0 85GRID 30 2.86919 3.9465 0.0 86GRID 31 3.08068 4.7358 0.0 87GRID 32 3.29217 5.5251 0.0 88GRID 33 2.07055 0.0 0.0

PAGE B U L K ..

7

..

D A T A 8

5.52510 0.0 9 .041 10 .041 11 .041 12 .041 13 .041 14 .041 15 .041 17 18 19 20 21 22 23 25 26 27 28 29 30 31 33 0.0 34 0.0 35 0.0 36 0.0 37 0.0 38 0.0 39 0.0 4

.11111 L

.10000

..

10

E C H O 9

..

10

.

1 +CA101 2.07055 +M00000 +M00001 +M00002 +M00003 +M00004 +M00005 +M00006

+M00007 +M00008 +M00009 +M00010 +M00011 +M00012 +M00013 +ER DENSITY MACH NO RFREQ

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7-57

Flutter Analysis Sample Problems

Chapter 7

919293949596979899100101102103104105106107108109110111112113114115116-

GRID GRID GRID GRID GRID MAT1 MKAERO1 +MK PAERO1 PARAM PARAM PARAM PARAM PARAM PARAM PARAM PSHELL SET1 +S1 +S2 SPC1 SPC1 SPC1 SPLINE1 TABDMP1 +T2000

36 2.70502 2.3679 37 2.91652 3.1572 38 3.12801 3.9465 39 3.3395 4.7358 40 3.55099 5.5251 1 9.2418+63.4993+6 0.45 .001 0.10 0.12 0.14 1 COUPMASS1 GRDPNT 17 KDAMP -1 LMODES 4 OPPHIPA 1 VREF 12.0 WTMASS .0025901 1 1 .041 1 100 2 4 6 15 18 20 22 31 34 36 38 1 6 1 THRU 1 12345 9 1 12345 25 100 101 101 124 2000 0.0 0.01 1000.0 0.01 ENDDATA TOTAL COUNT= 117

0.0 0.0 0.0 0.0 0.0 0.097464 +MK 0.16

8 24 40 40 100

0.20

1 9 25

11 27

13 29

+S1 +S2

.0 +T2000

ENDT

Listing 7-15. Sorted Bulk Data Entries for the 15-Degree Sweptback Wing by KE-Method EXAMPLE HA145E: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 14 KE-METHOD FLUTTER ANALYSIS, DOUBLET-LATTICE AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 17 M O * 4.000018E-02 -2.715799E-20 0.000000E+00 0.000000E+00 0.000000E+00 -1.105025E-01 * -2.715799E-20 4.000018E-02 0.000000E+00 0.000000E+00 0.000000E+00 2.960875E-02 * 0.000000E+00 0.000000E+00 4.000018E-02 1.105025E-01 -2.960875E-02 0.000000E+00 * 0.000000E+00 0.000000E+00 1.105025E-01 4.070250E-01 -1.090611E-01 0.000000E+00 * 0.000000E+00 0.000000E+00 -2.960875E-02 -1.090611E-01 4.038716E-02 0.000000E+00 * -1.105025E-01 2.960875E-02 0.000000E+00 0.000000E+00 0.000000E+00 4.474122E-01 S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 4.000018E-02 0.000000E+00 2.762550E+00 0.000000E+00 Y 4.000018E-02 7.402154E-01 0.000000E+00 0.000000E+00 Z 4.000018E-02 7.402154E-01 2.762550E+00 0.000000E+00 I(S) * 1.017562E-01 2.726544E-02 0.000000E+00 * * 2.726544E-02 1.847031E-02 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.202265E-01 * I(Q) * 1.098882E-01 * * 1.033837E-02 * * 1.202265E-01 * Q * 9.582864E-01 2.858097E-01 0.000000E+00 * * -2.858097E-01 9.582864E-01 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 1 4.656516E+04 2.157896E+02 3.434399E+01 2.485460E-05 1.157358E+00 2 2 1.741004E+06 1.319471E+03 2.100004E+02 9.088139E-06 1.582249E+01 3 3 2.677549E+06 1.636322E+03 2.604287E+02 8.523229E-06 2.282136E+01 4 4 1.590669E+07 3.988319E+03 6.347607E+02 7.943937E-06 1.263618E+02 5 5 2.676045E+07 5.173050E+03 8.233165E+02 0.0 0.0 6 6 6.705251E+07 8.188560E+03 1.303250E+03 0.0 0.0 7 7 1.020573E+08 1.010234E+04 1.607838E+03 0.0 0.0 8 8 2.060991E+08 1.435615E+04 2.284852E+03 0.0 0.0 9 9 2.839795E+08 1.685169E+04 2.682030E+03 0.0 0.0 10 10 3.717569E+08 1.928100E+04 3.068666E+03 0.0 0.0 11 11 4.330577E+08 2.081004E+04 3.312021E+03 0.0 0.0 12 12 5.752655E+08 2.398469E+04 3.817282E+03 0.0 0.0 13 13 6.478299E+08 2.545250E+04 4.050891E+03 0.0 0.0 14 14 7.371569E+08 2.715063E+04 4.321158E+03 0.0 0.0 15 15 1.062141E+09 3.259050E+04 5.186939E+03 0.0 0.0 16 16 1.340560E+09 3.661366E+04 5.827244E+03 0.0 0.0 17 17 1.504211E+09 3.878416E+04 6.172690E+03 0.0 0.0 18 18 2.016247E+09 4.490264E+04 7.146477E+03 0.0 0.0 19 19 2.896543E+09 5.381954E+04 8.565646E+03 0.0 0.0 20 20 3.734665E+09 6.111191E+04 9.726263E+03 0.0 0.0 21 21 3.846094E+09 6.201689E+04 9.870294E+03 0.0 0.0

7-58

NX Nastran Aeroelastic Analysis User’s Guide

* * * * * *

Flutter Analysis Sample Problems

22 24 4.477241E+09 23 25 4.639453E+09 24 27 4.942807E+09 25 31 5.256085E+09 26 35 5.759771E+09 27 37 6.100986E+09 28 36 6.657555E+09 29 34 7.332474E+09 30 33 8.441385E+09 31 32 9.166367E+09 32 30 1.070424E+10 33 29 1.173995E+10 34 28 1.259991E+10 35 26 1.455259E+10 36 23 1.897902E+10 37 22 2.018797E+10 EIGENVALUE = 4.656516E+04 CYCLES = 3.434399E+01 POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 10 G 11 G 12 G 13 G 14 G 15 G 16 G 18 G 19 G 20 G 21 G 22 G 23 G 24 G 26 G 27 G 28 G 29 G 30 G 31 G 32 G 33 G 34 G 35 G 36 G 37 G 38 G 39 G 40 G EIGENVALUE = 1.741004E+06 CYCLES = 2.100004E+02 POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 10 G 11 G 12 G 13 G 14 G 15 G 16 G 18 G 19 G 20 G 21 G 22 G 23 G 24 G 26 G 27 G 28 G 29 G 30 G 31 G 32 G 33 G 34 G 35 G

6.691219E+04 6.811353E+04 7.030509E+04 7.249886E+04 7.589316E+04 7.810880E+04 8.159384E+04 8.562987E+04 9.187702E+04 9.574114E+04 1.034613E+05 1.083510E+05 1.122493E+05 1.206341E+05 1.377644E+05 1.420844E+05

1.064941E+04 1.084060E+04 1.118940E+04 1.153855E+04 1.207877E+04 1.243140E+04 1.298606E+04 1.362842E+04 1.462268E+04 1.523768E+04 1.646638E+04 1.724460E+04 1.786503E+04 1.919951E+04 2.192588E+04 2.261343E+04

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R E A L T2

E I G E N V E C T O R T3 -3.320767E-03 3.354167E-02 1.197589E-01 2.415488E-01 3.895444E-01 5.599717E-01 7.364755E-01 9.197437E-01 3.985383E-02 1.283573E-01 2.494006E-01 4.009323E-01 5.693349E-01 7.477077E-01 9.293225E-01 5.363604E-02 1.427034E-01 2.734128E-01 4.279284E-01 5.990526E-01 7.780458E-01 9.601672E-01 4.663620E-02 1.527655E-01 2.906153E-01 4.520696E-01 6.267195E-01 8.077030E-01 9.904701E-01 -3.744976E-03 4.227901E-02 1.548405E-01 2.953975E-01 4.598230E-01 6.353986E-01 8.179794E-01 1.000000E+00

N O . R1

1

R E A L T2

E I G E N V E C T O R T3 1.107332E-02 -1.414919E-01 -4.007863E-01 -6.050646E-01 -6.624407E-01 -5.821251E-01 -3.570982E-01 -7.326151E-02 -1.446287E-01 -3.784626E-01 -5.368558E-01 -5.745575E-01 -4.620206E-01 -2.318033E-01 6.338773E-02 -1.403477E-01 -2.714325E-01 -3.485831E-01 -2.981144E-01 -1.210224E-01 1.500928E-01 4.641854E-01 -6.794433E-02 -1.554098E-01 -1.528570E-01 -2.636064E-02 2.173962E-01 5.304146E-01 8.653485E-01 1.004412E-02 -3.825526E-02 -1.152495E-01

N O . R1

2

R2

R3

R2

R3

NX Nastran Aeroelastic Analysis User’s Guide

7-59

Chapter 7

Flutter Analysis Sample Problems

36 G 37 G 38 G 39 G 40 G EIGENVALUE = 2.677549E+06 CYCLES = 2.604287E+02 POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 10 G 11 G 12 G 13 G 14 G 15 G 16 G 18 G 19 G 20 G 21 G 22 G 23 G 24 G 26 G 27 G 28 G 29 G 30 G 31 G 32 G 33 G 34 G 35 G 36 G 37 G 38 G 39 G 40 G EIGENVALUE = 1.590669E+07 CYCLES = 6.347607E+02 POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 10 G 11 G 12 G 13 G 14 G 15 G 16 G 18 G 19 G 20 G 21 G 22 G 23 G 24 G 26 G 27 G 28 G 29 G 30 G 31 G 32 G 33 G 34 G 35 G 36 G 37 G 38 G 39 G 40 G FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.4500 KFREQ 1./KFREQ VELOCITY 0.2000 5.0000000E+00 6.7383588E+02 0.1667 5.9998803E+00 8.1385651E+02 0.1429 6.9998603E+00 9.6011792E+02

7-60

-8.781217E-02 6.338255E-02 3.312269E-01 6.570764E-01 1.000000E+00 R E A L T2

E I G E N V E C T O R T3 2.114405E-03 1.986178E-02 6.028288E-02 1.323192E-01 2.601770E-01 4.728811E-01 7.355465E-01 1.000000E+00 2.551326E-03 1.791453E-02 5.546072E-02 1.576512E-01 3.465791E-01 5.893692E-01 8.437764E-01 -5.833367E-02 -1.493534E-01 -2.188800E-01 -1.930001E-01 -6.678208E-02 1.327620E-01 3.624148E-01 -1.448625E-01 -3.904501E-01 -5.610464E-01 -6.075832E-01 -5.242109E-01 -3.486537E-01 -1.321460E-01 -2.928281E-03 -1.885662E-01 -4.888113E-01 -6.918882E-01 -7.600891E-01 -6.841900E-01 -5.152062E-01 -2.968765E-01

N O . R1

3

R E A L T2

E I G E N V E C T O R T3 -2.639108E-02 3.380127E-01 7.191586E-01 6.149204E-01 8.150127E-02 -3.522899E-01 -4.111290E-01 -5.156524E-02 3.281422E-01 6.388384E-01 4.704424E-01 1.107625E-02 -3.550788E-01 -3.136066E-01 7.111400E-02 2.931411E-01 3.630863E-01 1.505998E-01 -2.019493E-01 -3.253174E-01 -6.143166E-02 4.675871E-01 1.313829E-01 1.363093E-01 -1.371148E-01 -3.879782E-01 -2.929345E-01 1.947475E-01 8.673875E-01 -1.393048E-02 8.065992E-02 6.916270E-02 -2.318921E-01 -4.519431E-01 -2.835256E-01 2.888194E-01 1.000000E+00

N O . R1

4

DENSITY RATIO = DAMPING -8.0309570E-02 -9.6444093E-02 -1.1731868E-01

NX Nastran Aeroelastic Analysis User’s Guide

9.6700E-01 FREQUENCY 2.4861040E+02 2.5023061E+02 2.5302904E+02

METHOD = KE COMPLEX -3.2339059E+02 -4.6823199E+02 -6.7008716E+02

R2

R3

R2

R3

EIGENVALUE 8.0665645E+03 9.7324424E+03 1.1462539E+04

Flutter Analysis Sample Problems

0.1250 0.1111 0.1000 POINT = KFREQ 0.2000 0.1667 0.1429 0.1250 0.1111 0.1000 POINT = KFREQ 0.2000 0.1667 0.1429 0.1250 0.1111 0.1000 POINT = KFREQ 0.2000 0.1667 0.1429 0.1250 0.1111 0.1000

8.0000000E+00 1.1146493E+03 9.0000896E+00 1.2800532E+03 1.0000000E+01 1.4570992E+03 2 MACH NUMBER = 0.4500 1./KFREQ VELOCITY 5.0000000E+00 4.2033994E+02 5.9998803E+00 4.5330319E+02 6.9998603E+00 4.7559656E+02 8.0000000E+00 4.9039053E+02 9.0000896E+00 4.9992407E+02 1.0000000E+01 5.0586630E+02 3 MACH NUMBER = 0.4500 1./KFREQ VELOCITY 5.0000000E+00 1.4238643E+03 5.9998803E+00 1.5988702E+03 6.9998603E+00 1.7381207E+03 8.0000000E+00 1.8456505E+03 9.0000896E+00 1.9268026E+03 1.0000000E+01 1.9934751E+03 4 MACH NUMBER = 0.4500 1./KFREQ VELOCITY 5.0000000E+00 9.3989601E+01 5.9998803E+00 1.1353393E+02 6.9998603E+00 1.3351437E+02 8.0000000E+00 1.5403619E+02 9.0000896E+00 1.7522746E+02 1.0000000E+01 1.9723370E+02

-1.4525376E-01 2.5702975E+02 -1.8538663E-01 2.6237131E+02 -2.4710195E-01 2.6879694E+02 DENSITY RATIO = 9.6700E-01 DAMPING FREQUENCY -7.3245011E-02 1.5508357E+02 -5.6584969E-02 1.3937387E+02 -3.1927105E-02 1.2533850E+02 -1.4126945E-03 1.1308038E+02 3.3927854E-02 1.0246896E+02 7.4329272E-02 9.3319183E+01 DENSITY RATIO = 9.6700E-01 DAMPING FREQUENCY -3.9331403E-02 5.2533191E+02 -2.8735427E-02 4.9159314E+02 -1.1333041E-02 4.5806354E+02 1.4551986E-02 4.2559317E+02 5.3239781E-02 3.9493491E+02 1.1295156E-01 3.6774432E+02 DENSITY RATIO = 9.6700E-01 DAMPING FREQUENCY -1.0171605E-01 3.4677273E+01 -1.2407344E-01 3.4907459E+01 -1.4841494E-01 3.5186317E+01 -1.7534544E-01 3.5519588E+01 -2.0578188E-01 3.5916210E+01 -2.4094324E-01 3.6384491E+01

-9.5884106E+02 1.3271563E+04 -1.3940487E+03 1.5167494E+04 -2.0816389E+03 1.7101801E+04 METHOD = KE COMPLEX EIGENVALUE -1.8411005E+02 5.0339712E+03 -1.5359367E+02 5.4331221E+03 -9.1048523E+01 5.7049785E+03 -4.1566267E+00 5.8846816E+03 1.0169495E+02 5.9965015E+03 2.2482840E+02 6.0578696E+03 METHOD = KE COMPLEX EIGENVALUE -3.3569098E+02 1.7076471E+04 -2.7552316E+02 1.9180506E+04 -1.1817968E+02 2.0856445E+04 1.6112596E+02 2.2146047E+04 6.1440735E+02 2.3097105E+04 1.3403313E+03 2.3808307E+04 METHOD = KE COMPLEX EIGENVALUE -5.6993584E+01 1.1235320E+03 -8.3715805E+01 1.3546294E+03 -1.1728425E+02 1.5891471E+03 -1.5901660E+02 1.8275884E+03 -2.1081007E+02 2.0703347E+03 -2.7523788E+02 2.3173606E+03

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Listing 7-16. Output Data for the 15-Degree Sweptback Wing by KE-Method

7.7 Low Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using Mach Box Aerodynamics (Example HA145F) Wind tunnel flutter tests of untapered aluminum plate wings with 15 degrees of sweep were reported by Tuovila and McCarty (1955). One supersonic test Mach number was m = 1.3, and this case is chosen to illustrate the use of the Mach Box aerodynamic method. A similar aluminum wing was tested at a subsonic Mach number of m = 0.45 and was the subject of the previous Example HA145E. The present example uses the same adjusted structural model as in Example HA145E because both the model weights and second vibration frequencies are identical and the same structural damping coefficient of g = 0.01 has been assumed. The PK-method of flutter analysis will be used to predict the flutter speed and frequency at the test density and Mach number. The new aerodynamic data begins with the AERO entry that specifies the reference parameters and symmetry conditions for the flutter analysis; the reference chord is 2.07055 in, sea-level density is chosen, and symmetry about the wind tunnel wall is assumed. Mach Box aerodynamics are specified on the CAERO3 and PAERO3 and related entries. The CAERO3 entry specifies certain intermediate sets of planform coordinates for splining on the wing and two control surfaces in its LISTW, LISTC1 and LISTC2 fields by reference to AEFACT entries, and its planform geometry on the continuation entry. These intermediate coordinates are selected at points that are representative of the modal deflections that determine the supersonic aerodynamic generalized forces. These points are interconnected with specified structural grid points by a user-selected spline. These points are also automatically interconnected (within the AMG module) by a surface spline to the Mach box centers that move with the Mach number. In this example LISTW on the CAERO3 101 entry refers to the AEFACT 11 entry, which lists 20 (x,y) pairs of coordinates. There are no control surfaces. The 20 (x,y) pairs are selected from the set of structural grid points for convenience, and since they are somewhat arbitrary, the grid point coordinates are rounded off to two decimals. The entry PAERO3 1 defines the number of chordwise Mach boxes between the upstream and downstream extremes of the planform, in this case 11. The user-specified spline to interconnect the Mach Box (x,y) pairs to the structural grid points is the surface spline SPLINE1 100. It connects the region specified by the range BOX1 = 101 through BOX2 = 120 on CAERO3 101 to the set of 20 structural grid points SETG = 100 listed on SET1 100. BOXi is a misnomer in the Mach Box method since it only refers to the ID of an (x,y) pair. BOX1 = 101 through BOX2 = 120 includes all of the (x,y) pairs on the AEFACT 11 entry. The MKAERO1 entry prescribes the aerodynamic database at Mach 1.3 and k = 0.03, 0.04, and 0.05 for subsequent interpolation in the PK-flutter analysis to include the experimental flutter reduced frequency, which was 0.0432. The vibration analysis is performed via EIGR 10 that requests the Modified Givens Eigenvalue Extraction method (MGlV) and six modes to be found and normalized on their maximum components. PARAM,OPPHIPA,1, along with a DISP request in the Case Control Section, provides the vibration modes at the structural and aerodynamic grid points in the output. The PARAM,LMODES entry specifies that the four lowest frequencies of the six vibration modes be used in the flutter analysis. The FLUTTER entry specifies the PK-method of analysis at the test density ratio on FLFACT 1, the test Mach number on FLFACT 2, and for the series of velocities on FLFACT 3 entries. Since no negative velocity is given on the FLFACT 3 entry, no flutter modes will be calculated. The FLUTTER entry also specifies three flutter solutions

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be output from the four mode formulation. The PARAM,VREF = 12.0 converts the analysis units of velocity from in/s to ft/s in the Flutter Summary. The ENDDATA entry completes the Bulk Data Section. Case Control Commands The Case Control Section begins with three title entries. ECHO = BOTH lists both the annotated and sorted Bulk Data entries. SPC = 1 provides the restraints on the wing root and on the plate element in-plane rotations. SDAMP = 2000 invokes the tabulated damping data. METHOD = 10 selects the method of vibration analysis, and FLUTTER = 30 chooses the method of flutter analysis. The DlSP = ALL entry (and PARAM,OPPHIPA,1) provides the modal displacements at both the structural and aerodynamic grid points. The OUTPUT(PLOT) entry identifies the plot packet for the structure plotter. The spacing in the title is specified by CSCALE 2.0. The CQUAD4 structural elements will be plotted. The FIND entry prescribes the origin and scales of the plots. The PLOT entry requests that the vibration modes be overlaid on the undeformed structure. The OUTPUT(XYOUT) entry identifies the plot packet for xy-plots. The first several statements are optional and define the plot frame and plot titles. The xy-plot request is for the first three flutter modes. Plots of damping on the top frame and frequency on the bottom frame, both versus velocity, will be plotted. BEGIN BULK concludes the Case Control Section. The problem identification ID NXN, HA145F begins the Executive Control Section. TIME 5 restricts the CPU time to 5.0 minutes. SOL 145 calls for the Structured Aerodynamic Flutter DMAP sequence. CEND completes the Executive Control Section. Output The input data are shown in Listing 7-17, the sorted Bulk Data entries are in Listing 7-18, and the representative output data follow in Listing 7-19. The results shown are discussed below. The output begins with the vibration results, which are given for the six modes requested. The results agree with those in the previous Example HA145E where four modes were requested. The vibration modes are not shown because they are identical with Example HA145E. The next two items of output are a summary of the planform geometry data for the points and lines shown in the description of the PAERO3 and a graphic display of the Mach boxes on and off the planform for the Mach number under consideration. At other Mach numbers the distribution of Mach boxes will be different, and the display provides a guide to indicate if a sufficient number of Mach boxes are on the main and control surfaces. The Flutter Summaries for the three flutter modes requested complete the output shown. The flutter damping and frequency versus velocity curves for the second mode (Point 2) are plotted in Figure 7-10 along with the experimental data. The first and third modes are stable while the second mode goes unstable at Vf = 1582 ft/s with a frequency of ff = 128 Hz for the four mode solution. The test results were Vf = 1280 ft/s and ff = 102 Hz. The reason for the unconservative prediction here is not known; the Mach Box method has been shown to be conservative in another application by Pendleton, French, and Noll (1987). There is, however, an unconservative aspect of linearized aerodynamic theory at supersonic speeds that must be mentioned. A consequence of neglecting thickness effects at supersonic speeds is an aft location of the aerodynamic center. These effects of thickness are included in Piston Theory but that theory is only valid at high Mach numbers. The correction matrix (WKK) in Eq. 1-21 might be used to adjust the Mach Box pressure solution so that the line of local aerodynamic centers moves forward by an amount determined by some appropriate aerodynamic method for thick wings in steady supersonic flow. The correction matrix might compensate for this unconservative feature of the Mach Box method. However, in this example the thin plate airfoil has only 2% thickness, and thickness effects alone do not explain the magnitude of the discrepancy between the predicted and measured flutter speeds observed here.

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Figure 7-10. Low Supersonic Flutter Damping and Frequency Curves for Sweptback Wing Model by Mach Box Method NX Nastran Aeroelastic Analysis User’s Guide 7-64 N A S T R A N E X E C U T I V E C O N T R O L

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$

$ POINT NO. $ $ SID C G1 G2 ETC SPC1 1 12345 9 SPC1 1 12345 25 SPC1 1 6 1 THRU 40 $ $ THE ASET1 ENTRY DEFINES DOFS THAT THE USER DESIRES TO BE $ INDEPENDENT. IT LISTS THE DOF COMPONENT NUMBERS AND THE $ GRID ID NO.S. $ $ V ID1 THRU ID2 ASET1 3 1 THRU 8 ASET1 3 10 THRU 24 ASET1 3 26 THRU 40 $ $ * STRUCTURAL DAMPING * $ $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUDING $ STRUCTURAL DAMPING IN THE EQUATIONS OF MOTION (SEE THE $ HANDBOOK FOR DYNAMIC ANALYSIS, SECTION 3.2.2). IF SET TO $ -1, MODAL STRUCTURAL DAMPING IS INCLUDED AS THE IMAGINARY $ PART OF A COMPLEX STIFFNESS MATRIX. $ $ N V1 V2 PARAM KDAMP +1 $ $ THE TABDMP1 ENTRY AND TABLE DEFINE MODAL DAMPING AS A $ TABULAR FUNCTION OF FREQUENCY. THE DAMPING VALUES ARE $ LINEAR BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE $ ZERO OUTSIDE THE TABULATED FREQUENCY RANGE. $ $ ID TABDMP1 2000 $ F1 G1 F2 G2 ETC ENDT +T2000 0.0 0.01 1000.0 0.01 ENDT $ * * * AERODYNAMIC DATA * * * $ $ (LB-IN-SEC SYSTEM) $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD $ AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=0 INDICATES $ THAT THE MODEL IS MOUNTED AS AN ISOLATED WING; SYMXY=0 $ INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH FROM THE $ FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 0 2.07055 1.145-7 1 $ THE CAERO3 ENTRY DEFINES PANEL ELEMENTS FOR THE MACH BOX $ METHOD: PID REFERS TO PROPERTIES DEFINED ON A PAERO3 $ ENTRY, CP REFERS TO THE COORDINATE SYSTEM OF POINTS 1 AND 4, $ LISTW IS THE ID OF AN AEFACT ENTRY ON WHICH ARE LISTED THE $ (X,Y) COORDINATES OF AN INTERMEDIATE SET OF POINTS ON THE $ WING, OPTIMALLY, THE MINIMUM NUMBER REQUIRED TO DEFINE THE $ MODE SHAPES AND CHORDWISE SLOPES OF ALL THE MODES USED IN $ THE ANALYSIS. THE INTERMEDIATE SET OF POINTS AND DISPLACE$ MENTS IS THEN USED TO GENERATE INTERPOLATED DOWNWASHES AT $ ANY REQUIRED MACH NUMBER. LISTC1 AND LISTC2 ARE CORRES$ PONDING AEFACT ENTRIES WHEN CONTROL SURFACES ARE PRESENT. $ THE CONTINUATION ENTRY LISTS THE COORDINATES AND CHORD AT $ POINTS 1 AND 4. $ EID PID CP LISTW LISTC1 LISTC2 CAERO3 101 1 0 11 $ X1 Y1 Z1 X12 X4 Y4 Z4 X43 +CA101 0.0 0.0 0.0 2.07055 1.48044 5.5251 0.0 2.07055 $ $ THE NUMBER OF X,Y PAIRS ON THE AEFACT ENTRY MUST EQUAL THE $ NUMBER OF POINTS SPECIFIED IN FIELDS 4 AND 5 OF ITS ASSOCIATED $ SPLINE ENTRY (IN THIS CASE SPLINE1,100). AEFACT 11 0.21 0.79 0.63 2.37 1.06 3.95 1.48 +AEF1 5.53 0.26 0.0 0.68 1.58 1.10 3.16 1.53 +AEF2 4.74 1.25 0.79 1.67 2.37 2.09 3.95 2.52 +AEF3 5.53 1.81 0.0 2.23 1.58 2.66 3.16 3.08 +AEF4 4.74 2.28 0.79 2.71 2.37 3.13 3.95 3.55 +AEF5 5.53 $ $ THE PAERO3 ENTRY DEFINES THE NUMBER OF CHORDWISE BOXES $ BETWEEN THE FOREMOST AND AFTMOST POINTS OF THE CONFIGUR$ ATION AND THE NUMBER OF CONTROL SURFACES. IN THE CASE WHEN $ CONTROL SURFACES ARE PRESENT THE COORDINATES OF THEIR $ CORNERS ARE LISTED. $ PID NBOX NCTRL PAERO3 1 11 0 $ * SURFACE SPLINE FIT ON THE WING * $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $ GRID POINTS ON THE SETG ENTRY TO THE REGION DEFINED BY $ AERODYNAMIC BOXES 101 THRU 109 OF THE REGION ON THE $ CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE $ SPLINE IS TO BE IMPOSED. $ $ IN SUPERSONIC CASES, BOX1 AND BOX2 ARE MISNOMERS; INSTEAD,

NX Nastran Aeroelastic Analysis User’s Guide

$ $

$ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +TDP +T2000 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CA101 $ $ $ $ +AEF1 +AEF2 +AEF3 +AEF4 +AEF5 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

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$

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$ THE DATA SPECIFY FIRST AND LAST POINT NUMBERS GENERATED BY $ THE CAERO3 ENTRY, THE FIRST NUMBER BEING THE ID NUMBER OF THE $ CAERO3 ENTRY AND THE LAST ONE BEING CONSISTENT WITH SEQUENTIAL $ NUMBERING OF THE X,Y PAIRS SPECIFIED BY THE CAERO3 ENTRY. $ THE SPLINE DETERMINED FROM EACH SETG IS USED TO GENERATE $ DEFLECTIONS AT THESE GIVEN INTERMEDIATE POINTS. $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 100 101 101 120 100 0.0 $ $ THE SET1 ENTRY DEFINES THE SET OF STRUCTURAL GRID POINTS TO $ BE USED BY THE SURFACE SPLINE FOR INTERPOLATION. $ SID G1 G2 ETC SET1 100 2 4 6 8 9 11 13 +S1 15 18 20 22 24 25 27 29 +S2 31 34 36 38 40 $ * * AERODYNAMIC DATABASE * * $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ TO GENERATE GENERALIZED AERO FORCE MATRICES. $ $ M1 M2 M3 ETC MKAERO1 1.3 $ K1 K2 K3 K4 ETC +MK .03 .04 .05 $ $ * * VIBRATION ANALYSIS * * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE. IN THIS CASE THE $ MODIFIED GIVENS METHOD. THREE MODES ARE DESIRED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 NO EIGR 10 MGIV 6 $ NORM +ER MAX $ OPPHIPA = 1 (WHEN ACCOMPANIED BY A DISPLACEMENT REQUEST IN $ THE CASE CONTROL DECK), WILL PRODUCE THE VIBRATION MODE $ DISPLACEMENTS IN THE OUTPUT. $ PARAM OPPHIPA 1 $ * * FLUTTER ANALYSIS * * $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. PARAM LMODES 4 $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE. $ $ SID METHOD DENS MACH RFRQ IMETH NVALUE EPS FLUTTER 30 PK 1 2 3 L 3 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 .20606 FLFACT 2 1.3 FLFACT 3 14400. 15600. 16800. 18000. 19200. 20400. $ $ THE PARAMETER VREF IS USED TO SCALE THE ANALYSIS UNITS OF $ VELOCITY, HERE IN/S, TO THE UNITS DESIRED IN THE OUTPUT $ FLUTTER SUMMARY TABLES, FT/S. $ PARAM VREF 12.0 $ ENDDATA INPUT BULK CARD ENTRY COUNT = 332

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$ $ $ $ $ $ $ $ $ +S1 +S2 $ $ $ $ $ +MK $ $ $ $ $ $ $ $ $ +ER $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO VELOCITY $ $ $ $ $ $

Flutter Analysis Sample Problems

Listing 7-17. Input Data for the 15-Degree Sweptback Wing with Mach Box Aerodynamics EXAMPLE HA145F: HALF SPAN 15-DEG SWEPT UNTAPERED WING OCTOBER 29, 2004 NX NASTRAN 10/28/04 PAGE 10 PK-METHOD OF FLUTTER ANALYSIS, MACH BOX AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES S O R T E D

66-

GRID

CARD COUNT 12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656 67686970717273747576777879-

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. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . AEFACT 11 0.21 0.79 0.63 2.37 1.06 3.95 1.48 +AEF1 +AEF1 5.53 0.26 0.0 0.68 1.58 1.10 3.16 1.53 +AEF2 +AEF2 4.74 1.25 0.79 1.67 2.37 2.09 3.95 2.52 +AEF3 +AEF3 5.53 1.81 0.0 2.23 1.58 2.66 3.16 3.08 +AEF4 +AEF4 4.74 2.28 0.79 2.71 2.37 3.13 3.95 3.55 +AEF5 +AEF5 5.53 AERO 0 2.07055 1.145-7 1 ASET1 3 1 THRU 8 ASET1 3 10 THRU 24 ASET1 3 26 THRU 40 CAERO3 101 1 0 11 +CA101 +CA101 0.0 0.0 0.0 2.07055 1.48044 5.5251 0.0 2.07055 CQUAD4 1 1 1 2 10 9 +M00000 +M00000 0.0 0.0 .041 .041 CQUAD4 2 1 2 3 11 10 +M00001 +M00001 0.0 0.0 .041 .041 CQUAD4 3 1 3 4 12 11 +M00002 +M00002 0.0 0.0 .041 .041 CQUAD4 4 1 4 5 13 12 +M00003 +M00003 0.0 0.0 .041 .041 CQUAD4 5 1 5 6 14 13 +M00004 +M00004 0.0 0.0 .041 .041 CQUAD4 6 1 6 7 15 14 +M00005 +M00005 0.0 0.0 .041 .041 CQUAD4 7 1 7 8 16 15 +M00006 +M00006 0.0 0.0 .041 .041 CQUAD4 8 1 9 10 18 17 CQUAD4 9 1 10 11 19 18 CQUAD4 10 1 11 12 20 19 CQUAD4 11 1 12 13 21 20 CQUAD4 12 1 13 14 22 21 CQUAD4 13 1 14 15 23 22 CQUAD4 14 1 15 16 24 23 CQUAD4 15 1 17 18 26 25 CQUAD4 16 1 18 19 27 26 CQUAD4 17 1 19 20 28 27 CQUAD4 18 1 20 21 29 28 CQUAD4 19 1 21 22 30 29 CQUAD4 20 1 22 23 31 30 CQUAD4 21 1 23 24 32 31 CQUAD4 22 1 25 26 34 33 +M00007 +M00007 .041 .041 0.0 0.0 CQUAD4 23 1 26 27 35 34 +M00008 +M00008 .041 .041 0.0 0.0 CQUAD4 24 1 27 28 36 35 +M00009 +M00009 .041 .041 0.0 0.0 CQUAD4 25 1 28 29 37 36 +M00010 +M00010 .041 .041 0.0 0.0 CQUAD4 26 1 29 30 38 37 +M00011 +M00011 .041 .041 0.0 0.0 CQUAD4 27 1 30 31 39 38 +M00012 +M00012 .041 .041 0.0 0.0 CQUAD4 28 1 31 32 40 39 +M00013 +M00013 .041 .041 0.0 0.0 EIGR 10 MGIV 6 +ER +ER MAX FLFACT 1 .20606 DENSITY FLFACT 2 1.3 MACH NO FLFACT 3 14400. 15600. 16800. 18000. 19200. 20400. VELOCITY FLUTTER 30 PK 1 2 3 L 3 GRID 1 0.0 0.0 0.0 GRID 2 .211491 .7893 0.0 GRID 3 .422983 1.5786 0.0 GRID 4 .634474 2.3679 0.0 GRID 5 .845966 3.1572 0.0 1.05746 3.9465 0.0 GRID 7 1.26895 4.7358 0.0 GRID 8 1.48044 5.5251 0.0 GRID 9 .258819 0.0 0.0 GRID 10 .47031 .7893 0.0 GRID 11 .681802 1.5786 0.0 GRID 12 .893293 2.3679 0.0 GRID 13 1.10478 3.1572 0.0 GRID 14 1.31628 3.9465 0.0 GRID 15 1.52777 4.7358 0.0 GRID 16 1.73926 5.5251 0.0 GRID 17 1.03528 0.0 0.0 GRID 18 1.24677 .7893 0.0 GRID 19 1.45826 1.5786 0.0

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8081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120-

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID MAT1 MKAERO1 +MK PAERO3 PARAM PARAM PARAM PARAM PARAM PARAM PSHELL SET1 +S1 +S2 SPC1 SPC1 SPC1 SPLINE1 TABDMP1 +T2000 ENDDATA TOTAL COUNT=

20 1.66975 2.3679 21 1.88124 3.1572 22 2.09273 3.9465 23 2.30422 4.7358 24 2.51572 5.5251 25 1.81173 0.0 26 2.02322 .7893 27 2.23471 1.5786 28 2.44621 2.3679 29 2.6577 3.1572 30 2.86919 3.9465 31 3.08068 4.7358 32 3.29217 5.5251 33 2.07055 0.0 34 2.28204 .7893 35 2.49353 1.5786 36 2.70502 2.3679 37 2.91652 3.1572 38 3.12801 3.9465 39 3.3395 4.7358 40 3.55099 5.5251 1 9.2418+63.4993+6 1.3 .03 .04 .05 1 11 0 COUPMASS1 KDAMP +1 LMODES 4 OPPHIPA 1 VREF 12.0 WTMASS .0025901 1 1 .041 1 100 2 4 6 15 18 20 22 31 34 36 38 1 6 1 THRU 1 12345 9 1 12345 25 100 101 101 120 2000 0.0 0.01 1000.0 0.01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.097464 +MK

8 24 40 40 100

1 9 25

11 27

13 29

+S1 +S2

0.0 +T2000

ENDT

121

Listing 7-18. Sorted Bulk Data for the 15-Degree Sweptback Wing with Mach Box Aerodynamics R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 1 4.656516E+04 2.157896E+02 3.434399E+01 2.485460E-05 1.157358E+00 2 2 1.741003E+06 1.319471E+03 2.100003E+02 9.088159E-06 1.582251E+01 3 3 2.677548E+06 1.636322E+03 2.604286E+02 8.523231E-06 2.282136E+01 4 4 1.590605E+07 3.988239E+03 6.347479E+02 7.944785E-06 1.263701E+02 5 5 2.676011E+07 5.173018E+03 8.233114E+02 8.134972E-06 2.176928E+02 6 6 6.703381E+07 8.187417E+03 1.303068E+03 4.671878E-06 3.131738E+02 7 7 1.020194E+08 1.010046E+04 1.607539E+03 0.0 0.0 8 8 2.059570E+08 1.435120E+04 2.284065E+03 0.0 0.0 9 9 2.830802E+08 1.682499E+04 2.677780E+03 0.0 0.0 10 10 3.717334E+08 1.928039E+04 3.068569E+03 0.0 0.0 SUPERSONIC MACH BOX PROGRAM CONTROL DATA F CNTRL2 F CNTRL1 F CRANK (LEADING EDGE) F CRANK (TRAILING EDGE) F ANTI-SYMMETRIC GEOMETRY DATA N X Y SWEEP ANGLE TANGENT 1 0.000000E+00 0.000000E+00 1-5 0.149999E+02 0.267948E+00 2 0.207055E+01 0.000000E+00 2-6 0.149999E+02 0.267948E+00 3 0.355099E+01 0.552510E+01 4 0.148044E+01 0.552510E+01 5 0.148044E+01 0.552510E+01 5-4 0.000000E+00 6 0.355099E+01 0.552510E+01 6-3 0.000000E+00 7 0.000000E+00 0.000000E+00 8 0.000000E+00 0.000000E+00 9 0.000000E+00 0.000000E+00 10 0.000000E+00 0.000000E+00 11 0.000000E+00 0.000000E+00 12 0.000000E+00 0.000000E+00 AREA OF MAIN (SEMISPAN) AREA OF CNTRL1 AREA OF CNTRL2 0.114400E+02 0.000000E+00 0.000000E+00 GRAPHIC DISPLAY OF REGIONS ON MAIN SEMISPAN MACH NUMBER 1.300 BOX WIDTH 0.184029 BOX LENGTH 0.152866 SSSS S MAIN SSSSSSS 1 CNTRL1 SSSSSSSSSS 2 CNTRL2 SSSSSSSSSSSSS . DIAPHRAGM SSSSSSSSSSSSSSS , WAKE SSSSSSSSSSSSSSS.

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SSSSSSSSSSSSSSS.. SSSSSSSSSSSSSS... SSSSSSSSSS... SSSSSSS.. SSSS. S POINT = KFREQ 0.0263 0.0266 0.0275 0.0292 0.0327 0.0324 POINT = KFREQ 0.0782 0.0689 0.0603 0.0517 0.0423 0.0373 POINT = KFREQ 0.1210 0.1123 0.1048 0.0984 0.0928 0.0878

1

MACH NUMBER = 1.3000 1./KFREQ VELOCITY 3.8016720E+01 1.2000000E+03 3.7556377E+01 1.3000000E+03 3.6366863E+01 1.4000000E+03 3.4202660E+01 1.5000000E+03 3.0585239E+01 1.6000000E+03 3.0884527E+01 1.7000000E+03 2 MACH NUMBER = 1.3000 1./KFREQ VELOCITY 1.2782905E+01 1.2000000E+03 1.4511147E+01 1.3000000E+03 1.6595564E+01 1.4000000E+03 1.9342052E+01 1.5000000E+03 2.3650806E+01 1.6000000E+03 2.6796661E+01 1.7000000E+03 3 MACH NUMBER = 1.3000 1./KFREQ VELOCITY 8.2624607E+00 1.2000000E+03 8.9064770E+00 1.3000000E+03 9.5409346E+00 1.4000000E+03 1.0165442E+01 1.5000000E+03 1.0779650E+01 1.6000000E+03 1.1383230E+01 1.7000000E+03

FLUTTER SUMMARY DENSITY RATIO = 2.0606E-01 DAMPING FREQUENCY -2.1929038E-01 5.8230736E+01 -2.3566487E-01 6.3856529E+01 -2.5580588E-01 7.1017899E+01 -2.8859755E-01 8.0905304E+01 -4.1954195E-01 9.6505875E+01 -7.8819412E-01 1.0154384E+02 DENSITY RATIO = 2.0606E-01 DAMPING FREQUENCY -9.1660082E-02 1.7317984E+02 -9.5254958E-02 1.6526743E+02 -9.4836451E-02 1.5562582E+02 -8.1455477E-02 1.4306529E+02 3.0586945E-02 1.2480145E+02 3.4935716E-01 1.1703448E+02 DENSITY RATIO = 2.0606E-01 DAMPING FREQUENCY -3.2511093E-02 2.6792761E+02 -3.4038611E-02 2.6926691E+02 -3.5570826E-02 2.7069656E+02 -3.7107173E-02 2.7221411E+02 -3.8647830E-02 2.7381735E+02 -4.0193569E-02 2.7550473E+02

METHOD = PK COMPLEX EIGENVALUE -4.0116379E+01 3.6587451E+02 -4.7277016E+01 4.0122241E+02 -5.7072674E+01 4.4621863E+02 -7.3353279E+01 5.0834305E+02 -1.2719764E+02 6.0636432E+02 -2.5144133E+02 6.3801880E+02 METHOD = PK COMPLEX EIGENVALUE -4.9868633E+01 1.0881211E+03 -4.9456654E+01 1.0384059E+03 -4.6366772E+01 9.7782593E+02 -3.6610401E+01 8.9890582E+02 1.1992387E+01 7.8415070E+02 1.2844978E+02 7.3534937E+02 METHOD = PK COMPLEX EIGENVALUE -2.7365219E+01 1.6834390E+03 -2.8794180E+01 1.6918540E+03 -3.0250084E+01 1.7008368E+03 -3.1733528E+01 1.7103717E+03 -3.3245739E+01 1.7204453E+03 -3.4788486E+01 1.7310474E+03

Listing 7-19. Output for the 15-Degree Sweptback Wing with Mach Box Aerodynamics

7.8 Low Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using ZONA51 Aerodynamics (Examples HA145FA and HA145FB) The previous Example HA145F illustrated the Mach Box method within its limitations to an isolated wing. The ZONA51 supersonic lifting surface theory is more general in that it can analyze multiple interfering surfaces, and it is illustrated here in this variation on HA145F designated HA145FA. It is necessary to replace the CAERO3 and PAERO3 entries in HA145F by CAERO1 and PAERO1 entries. For this application the CAERO1 entry is specified with 10 equal chordwise and 10 equal spanwise divisions. The SPLINE1 entry is also modified to connect aerodynamic boxes 101 through 200 to the same 20 grid points of the structure as in HA145F. The new input data are shown in Listing 7-20. The Sorted Bulk Data entries for the modified input are shown next in Listing 7-21 followed by the flutter solution summaries for the three output flutter modes in Listing 7-22. The second flutter mode (Point 2) is the critical one. Linear interpolation between tabulated values gives a flutter speed Vf = 1576 ft/s and a flutter frequency of ff = 132 Hz. These may be compared to the test results, Vf = 1280 ft/s and ff = 102 Hz, and also to the Mach Box results of Example HA145F, Vf = 1540 ft/s and ff = 134 Hz. At the end of the discussion of the Mach Box correlation in the previous section, it was noted that neglecting thickness effects in supersonic flutter analysis is unconservative, and that the correction matrix [Wkk ] (see Eq. 1-21) might be utilized to adjust the aerodynamic center predicted by the Mach Box method. The ZONA51 supersonic lifting surface theory also neglects thickness effects so a correction factor scheme is suggested here based on Piston Theory. This variation of HA145FA is called HA145FB. The oscillatory aerodynamic coefficients from Piston Theory have been developed by Rodden, Farkas, Malcom, and Kliszewski (1962) to include modifications from the second order of Van Dyke (1952) in order to extend the validity of Piston Theory to lower supersonic Mach numbers. The oscillatory coefficients are referred to the semichord, so the aerodynamic center as a fraction of the semichord measured from the leading edge is given by

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Equation 7-2.

Equation 7-3.

Equation 7-4. and are discussed on the CAERO5 Bulk Data entry, the third-order The coefficients coefficient is C3 = (γ + 1)/12 from Piston Theory. The thickness integrals I1 , I2 , I4 , and I5 are also discussed on the CAERO5 Bulk Data entry. Eq. 7-4 is conservative (i.e., predicts a farther forward aerodynamic center) if the third-order effect of trim angle of attack α0 is neglected. Therefore, Eq. 7-4 will be rewritten for the purposes of the correction as

Equation 7-5.

Figure 7-11. Linear Chordwise Correction Factor

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Figure 7-11 shows a linear variation in the chordwise correction factor required to shift forward the theoretical uniform pressure distribution due to the angle of attack of a flat plate airfoil. The linear correction factor is

Equation 7-6. where ξ is measured aft from the leading edge. If the chordwise load is I(ξ), the corrected lift is

Equation 7-7. and the moment about the leading edge is

Equation 7-8. The aerodynamic center is located at

Equation 7-9. Assume I(ξ) = 1, so L = 1.0, Mo = 1/2 − w/12, and the loading correction is found to be

Equation 7-10. and the correction factor is found from Eq. 7-6. The procedure is illustrated by modifying the above Example HA145FA. The thickness integrals can be obtained from Example HA75G (Rodden, 1987).

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The airfoil coefficients are

The example wing has a leading edge sweep angle of = 15° and air has a specific heat ratio of γ = 1.400, so the following coefficients, aerodynamic center locations, and loading corrections are found: M 1.3

1.6534

2.7502

0.2

xac /c

w

0.4626

0.4488

The ZONA51 idealization for the example wing uses 10 equal chordwise and 10 equal spanwise divisions. The correction factors are then the same on each strip and are tabulated in Table 7-1 from Eq. 7-6. Table 7-1. Correction Factors for the 15-Degree Sweptback Wing at Mach 1.3 ξ

W

0.05

1.2020

0.15

1.1571

0.25

1.1122

0.35

1.0673

0.45

1.0224

0.55

0.9776

0.65

0.9327

0.75

0.8878

0.85

0.8429

0.95

0.7980

Listing 7-23 shows the Input Data for HA145FB that includes the DMI input for the correction factors for the 10 strips. Listing 7-24 gives the sorted Bulk Data entries. The Flutter Summaries for the three output flutter modes are shown in Listing 7-25; interpolation between tabulated values for the critical mode gives a flutter speed Vf = 1405 ft/s and frequency ff = 129 Hz. These

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results are seen to compare somewhat more closely to the test results discussed above (Vf = 1280 ft/s and ff = 102 Hz. N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA145FA $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145FA $$$$$$$$ $ $ $ $ MODEL DESCRIPTION MODEL A OF NASA TN D-1824 $ $ HALF SPAN 15 DEGREE SWEPT WING $ $ 28 QUAD4 PANEL MODEL $ $ $ $ SOLUTION PK-FLUTTER ANALYSIS METHOD $ $ USING ZONA51 AERODYNAMICS $ $ AT MACH NO. 1.3 $ $ $ $ OUTPUT PLOTS OF VIBRATION MODES AND X-Y $ $ PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ $ TIME 5 $ CPU TIME IN MINUTES SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145FA: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 2 PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145FA: HALF SPAN 15-DEG SWEPT UNTAPERED WING 2 SUBT = PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO 3 LABEL = 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES 4 ECHO = BOTH 5 SPC = 1 $ 6 SDAMP = 2000 7 METHOD = 10 $ MODIFIED GIVENS 8 FMETHOD = 30 $ PK-FLUTTER METHOD 9 DISP = ALL $ PRINT VIBRATION MODES 10 OUTPUT(PLOT) 11 CSCALE 2.0 12 PLOTTER NASTRAN 13 SET 1 = QUAD4 14 PTITLE = STRUCTURAL ELEMENTS 15 FIND SCALE, ORIGIN 1, SET 1 16 PLOT MODAL 0 17 OUTPUT (XYOUT) 18 CSCALE 2.0 19 PLOTTER NASTRAN 20 CURVELINESYMBOL = -6 21 YTTITLE = DAMPING G 22 YBTITLE = FREQUENCY F HZ 23 XTITLE = VELOCITY V (IN/SEC) 24 XTGRID LINES = YES 25 XBGRID LINES = YES 26 YTGRID LINES = YES 27 YBGRID LINES = YES 28 UPPER TICS = -1 29 TRIGHT TICS = -1 30 BRIGHT TICS = -1 31 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 32 BEGIN BULK PAGE 3 EXAMPLE HA145FA: HALF SPAN 15-DEG SWEPT UNTAPERED WING PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES I N P U T B U L K D A T A D E C K E C H O 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . . 1 .. 2 .. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ INCLUDE PLATE_STRUCT.DAT $ $ $ * * MASS AND INERTIA PROPERTIES * * $ $ $ $ THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES FOR LINEAR, $ $ TEMPERATURE-INDEPENDENT, ISOTROPIC MATERIALS. IT LISTS $ $ YOUNGS MODULUS, THE SHEAR MODULUS, POISSONS RATIO, $ $ MASS DENSITY, THERMAL EXPANSION COEFFICIENT AND STRUCTURAL $ $ ELEMENT DAMPING. $ $ $ $ MID E G NU RHO A TREF GE MAT1 1 9.2418+63.4993+6 0.097464 $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $ $ THE SPC1 ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS. $

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$

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$ IT LISTS THE DOF COMPONENTS TO BE CONSTRAINED AND THE GRID $ POINT NO. $ $ SID C G1 G2 ETC SPC1 1 12345 9 SPC1 1 12345 25 SPC1 1 6 1 THRU 40 $ $ THE ASET1 ENTRY DEFINES DOFS THAT THE USER DESIRES TO BE $ INDEPENDENT. IT LISTS THE DOF COMPONENT NUMBERS AND THE $ GRID ID NO.S. $ $ V ID1 THRU ID2 ASET1 3 1 THRU 8 ASET1 3 10 THRU 24 ASET1 3 26 THRU 40 $ $ * STRUCTURAL DAMPING * $ $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUDING $ STRUCTURAL DAMPING IN THE EQUATIONS OF MOTION (SEE THE $ HANDBOOK FOR DYNAMIC ANALYSIS, SECTION 3.2.2). IF SET TO $ -1, MODAL STRUCTURAL DAMPING IS INCLUDED AS THE IMAGINARY $ PART OF A COMPLEX STIFFNESS MATRIX. $ $ N V1 V2 PARAM KDAMP +1 $ $ THE TABDMP1 ENTRY AND TABLE DEFINE MODAL DAMPING AS A $ TABULAR FUNCTION OF FREQUENCY. THE DAMPING VALUES ARE $ LINEAR BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE $ ZERO OUTSIDE THE TABULATED FREQUENCY RANGE. $ $ ID TABDMP1 2000 $ F1 G1 F2 G2 ETC ENDT +T2000 0.0 0.01 1000.0 0.01 ENDT $ * * * AERODYNAMIC DATA * * * $ $ $ (LB-IN-SEC SYSTEM) $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD $ AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=0 INDICATES $ THAT THE MODEL IS MOUNTED AS AN ISOLATED WING; SYMXY=0 $ INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH FROM THE $ FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 0 2.07055 1.145-7 1 $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET LATTICE AERODYNAMICS. $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS. $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ EXTRA POINT IDS. $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 101 1 0 10 10 1 $ ( FWD LEFT POINT ) ROOTCHORD ( FWD RIGHT POINT ) TIP CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1 $ $ * SURFACE SPLINE FIT ON THE WING * $ $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $ GRID POINTS ON THE SETG ENTRY TO THE REGION DEFINED BY $ AERODYNAMIC BOXES 101 THRU 200 OF THE REGION ON THE $ CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE $ SPLINE IS TO BE IMPOSED. $ $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 100 101 101 200 100 0.0 $ $ THE SET1 ENTRY DEFINES THE SET OF STRUCTURAL GRID POINTS TO

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$ $ $

$ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +TDP +T2000 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CA101

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

Flutter Analysis Sample Problems

$

$ BE USED BY THE SURFACE SPLINE FOR INTERPOLATION. $ $ SID G1 G2 ETC SET1 100 2 4 6 8 9 11 13 +S1 15 18 20 22 24 25 27 29 +S2 31 34 36 38 40 $ * * AERODYNAMIC DATABASE * * $ $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ TO GENERATE GENERALIZED AERO FORCE MATRICES. $ $ M1 M2 M3 ETC MKAERO1 1.3 $ K1 K2 K3 K4 ETC +MK .03 .04 .05 $ $ * * VIBRATION ANALYSIS * * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE. IN THIS CASE THE $ MODIFIED GIVENS METHOD. THREE MODES ARE DESIRED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 NO EIGR 10 MGIV 6 $ NORM +ER MAX $ $ OPPHIPA = 1 (WHEN ACCOMPANIED BY A DISPLACEMENT REQUEST IN $ THE CASE CONTROL DECK), WILL PRODUCE THE VIBRATION MODE $ DISPLACEMENTS IN THE OUTPUT. $ PARAM OPPHIPA 1 $ $ * * FLUTTER ANALYSIS * * $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. $ PARAM LMODES 4 $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE. $ $ SID METHOD DENS MACH RFRQ IMETH NVALUE EPS FLUTTER 30 PK 1 2 3 L 3 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 .20606 FLFACT 2 1.3 FLFACT 3 14400. 15600. 16800. 18000. 19200. 20400. $ $ THE PARAMETER VREF IS USED TO SCALE THE ANALYSIS UNITS OF $ VELOCITY, HERE IN/S, TO THE UNITS DESIRED IN THE OUTPUT $ FLUTTER SUMMARY TABLES, FT/S. $ PARAM VREF 12.0 $ ENDDATA INPUT BULK DATA CARD COUNT = 310

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$ $ +S1 +S2 $ $ $ $ $ $ +MK $ $ $ $ $ $ $ $ $ +ER $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO VELOCITY $ $ $ $ $ $

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Listing 7-20. Input Data for the 15-Degree Sweptback Wing with ZONA51 Aerodynamics EXAMPLE HA145F_A: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 10 PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AERO 0 2.07055 1.145-7 1 2ASET1 3 1 THRU 8 3ASET1 3 10 THRU 24 4ASET1 3 26 THRU 40 5CAERO1 101 1 0 10 10 1 +CA101 6+CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 7CQUAD4 1 1 1 2 10 9 +M00000 8+M00000 0.0 0.0 .041 .041 9CQUAD4 2 1 2 3 11 10 +M00001 10+M00001 0.0 0.0 .041 .041 11CQUAD4 3 1 3 4 12 11 +M00002 12+M00002 0.0 0.0 .041 .041 13CQUAD4 4 1 4 5 13 12 +M00003 14+M00003 0.0 0.0 .041 .041 15CQUAD4 5 1 5 6 14 13 +M00004 16+M00004 0.0 0.0 .041 .041 17CQUAD4 6 1 6 7 15 14 +M00005 18+M00005 0.0 0.0 .041 .041 19CQUAD4 7 1 7 8 16 15 +M00006 20+M00006 0.0 0.0 .041 .041 21CQUAD4 8 1 9 10 18 17 22CQUAD4 9 1 10 11 19 18 23CQUAD4 10 1 11 12 20 19 24CQUAD4 11 1 12 13 21 20 25CQUAD4 12 1 13 14 22 21 26CQUAD4 13 1 14 15 23 22 27CQUAD4 14 1 15 16 24 23 28CQUAD4 15 1 17 18 26 25 29CQUAD4 16 1 18 19 27 26 30CQUAD4 17 1 19 20 28 27 31CQUAD4 18 1 20 21 29 28 32CQUAD4 19 1 21 22 30 29 33CQUAD4 20 1 22 23 31 30 34CQUAD4 21 1 23 24 32 31 35CQUAD4 22 1 25 26 34 33 +M00007 36+M00007 .041 .041 0.0 0.0 37CQUAD4 23 1 26 27 35 34 +M00008 38+M00008 .041 .041 0.0 0.0 39CQUAD4 24 1 27 28 36 35 +M00009 40+M00009 .041 .041 0.0 0.0 41CQUAD4 25 1 28 29 37 36 +M00010 42+M00010 .041 .041 0.0 0.0 43CQUAD4 26 1 29 30 38 37 +M00011 44+M00011 .041 .041 0.0 0.0 45CQUAD4 27 1 30 31 39 38 +M00012 46+M00012 .041 .041 0.0 0.0 47CQUAD4 28 1 31 32 40 39 +M00013 48+M00013 .041 .041 0.0 0.0 49EIGR 10 MGIV 6 +ER 50+ER MAX 51FLFACT 1 .20606 DENSITY 52FLFACT 2 1.3 MACH NO 53FLFACT 3 14400. 15600. 16800. 18000. 19200. 20400. VELOCITY 54FLUTTER 30 PK 1 2 3 L 3 55GRID 1 0.0 0.0 0.0 56GRID 2 .211491 .7893 0.0 57GRID 3 .422983 1.5786 0.0 58GRID 4 .634474 2.3679 0.0 59GRID 5 .845966 3.1572 0.0 60GRID 6 1.05746 3.9465 0.0 61GRID 7 1.26895 4.7358 0.0 62GRID 8 1.48044 5.5251 0.0 63GRID 9 .258819 0.0 0.0 64GRID 10 .47031 .7893 0.0 65GRID 11 .681802 1.5786 0.0 66GRID 12 .893293 2.3679 0.0 67GRID 13 1.10478 3.1572 0.0 68GRID 14 1.31628 3.9465 0.0 69GRID 15 1.52777 4.7358 0.0 70GRID 16 1.73926 5.5251 0.0 71GRID 17 1.03528 0.0 0.0 72GRID 18 1.24677 .7893 0.0 73GRID 19 1.45826 1.5786 0.0 74GRID 20 1.66975 2.3679 0.0 75GRID 21 1.88124 3.1572 0.0 76GRID 22 2.09273 3.9465 0.0 77GRID 23 2.30422 4.7358 0.0 78GRID 24 2.51572 5.5251 0.0 79GRID 25 1.81173 0.0 0.0 80GRID 26 2.02322 .7893 0.0 81GRID 27 2.23471 1.5786 0.0

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828384858687888990919293949596979899100101102103104105106107108109110111112113114-

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID MAT1 MKAERO1 +MK PAERO1 PARAM PARAM PARAM PARAM PARAM PARAM PSHELL SET1 +S1 +S2 SPC1 SPC1 SPC1 SPLINE1 TABDMP1 +T2000 ENDDATA TOTAL COUNT=

28 2.44621 2.3679 29 2.6577 3.1572 30 2.86919 3.9465 31 3.08068 4.7358 32 3.29217 5.5251 33 2.07055 0.0 34 2.28204 .7893 35 2.49353 1.5786 36 2.70502 2.3679 37 2.91652 3.1572 38 3.12801 3.9465 39 3.3395 4.7358 40 3.55099 5.5251 1 9.2418+63.4993+6 1.3 .03 .04 .05 1 COUPMASS1 KDAMP +1 LMODES 4 OPPHIPA 1 VREF 12.0 WTMASS .0025901 1 1 .041 1 100 2 4 6 15 18 20 22 31 34 36 38 1 6 1 THRU 1 12345 9 1 12345 25 100 101 101 200 2000 0.0 0.01 1000.0 0.01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.097464 +MK

8 24 40 40 100

1 9 25

11 27

13 29

+S1 +S2

0.0 +T2000

ENDT

115

Listing 7-21. Sorted Bulk Data for the 15-Degree Sweptback Wing with ZONA51 Aerodynamics EXAMPLE HA145FA: HALF SPAN 15-DEG SWEPT UNTAPERED WING PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES FLUTTER SUMMARY POINT = 1 MACH NUMBER = 1.3000 DENSITY RATIO = 2.0606E-01 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.0264 3.7883450E+01 1.2000000E+03 -2.2285075E-01 5.8435585E+01 0.0267 3.7442520E+01 1.3000000E+03 -2.3879832E-01 6.4050705E+01 0.0276 3.6297466E+01 1.4000000E+03 -2.5805500E-01 7.1153679E+01 0.0292 3.4227066E+01 1.5000000E+03 -2.8823844E-01 8.0847618E+01 0.0326 3.0688646E+01 1.6000000E+03 -3.9644197E-01 9.6180687E+01 0.0328 3.0525984E+01 1.7000000E+03 -7.5888366E-01 1.0273653E+02 POINT = 2 MACH NUMBER = 1.3000 DENSITY RATIO = 2.0606E-01 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.0788 1.2696283E+01 1.2000000E+03 -9.2844881E-02 1.7436139E+02 0.0695 1.4390484E+01 1.3000000E+03 -9.7258307E-02 1.6665317E+02 0.0609 1.6422031E+01 1.4000000E+03 -9.8186642E-02 1.5727034E+02 0.0524 1.9070755E+01 1.5000000E+03 -8.8135026E-02 1.4510052E+02 0.0431 2.3212460E+01 1.6000000E+03 1.5437534E-03 1.2715823E+02 0.0377 2.6514278E+01 1.7000000E+03 3.2057297E-01 1.1828094E+02 POINT = 3 MACH NUMBER = 1.3000 DENSITY RATIO = 2.0606E-01 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.1210 8.2636967E+00 1.2000000E+03 -3.3892032E-02 2.6788754E+02 0.1123 8.9067831E+00 1.3000000E+03 -3.5415586E-02 2.6925769E+02 0.1048 9.5398874E+00 1.4000000E+03 -3.6938693E-02 2.7072629E+02 0.0984 1.0162581E+01 1.5000000E+03 -3.8461521E-02 2.7229074E+02 0.0928 1.0774477E+01 1.6000000E+03 -3.9984874E-02 2.7394882E+02 0.0879 1.1375218E+01 1.7000000E+03 -4.1509837E-02 2.7569876E+02

PAGE

43

METHOD = PK COMPLEX EIGENVALUE -4.0911121E+01 3.6716162E+02 -4.8051292E+01 4.0244247E+02 -5.7684547E+01 4.4707175E+02 -7.3209763E+01 5.0798059E+02 -1.1978912E+02 6.0432111E+02 -2.4493452E+02 6.4551270E+02 METHOD = PK COMPLEX EIGENVALUE -5.0857868E+01 1.0955449E+03 -5.0920208E+01 1.0471128E+03 -4.8511993E+01 9.8815869E+02 -4.0176067E+01 9.1169354E+02 6.1669767E-01 7.9895874E+02 1.1912189E+02 7.4318109E+02 METHOD = PK COMPLEX EIGENVALUE -2.8523315E+01 1.6831871E+03 -2.9957973E+01 1.6917960E+03 -3.1416792E+01 1.7010234E+03 -3.2901009E+01 1.7108533E+03 -3.4412407E+01 1.7212712E+03 -3.5953049E+01 1.7322665E+03

Listing 7-22. Output for the 15-Degree Sweptback Wing with ZONA51 Aerodynamics

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N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA145FB $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145FB $$$$$$$$ $ $ $ $ MODEL DESCRIPTION MODEL A OF NASA TN D-1824 $ $ HALF SPAN 15 DEGREE SWEPT WING $ $ 28 QUAD4 PANEL MODEL $ $ $ $ SOLUTION PK-FLUTTER ANALYSIS METHOD $ $ USING ZONA51 AERODYNAMICS $ $ AT MACH NO. 1.3 AND CORRECTIONS $ $ $ $ OUTPUT PLOTS OF VIBRATION MODES AND X-Y $ $ PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ $ TIME 5 $ CPU TIME IN MINUTES SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145FB: HALF SPAN 15-DEG SWEPT UNTAPERED PAGE 2 PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO WITH WKK 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145FB: HALF SPAN 15-DEG SWEPT UNTAPERED WING 2 SUBT = PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO WITH WKK 3 LABEL = 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES 4 ECHO = BOTH 5 SPC = 1 $ 6 SDAMP = 2000 7 METHOD = 10 $ MODIFIED GIVENS 8 FMETHOD = 30 $ PK-FLUTTER METHOD 9 DISP = ALL $ PRINT VIBRATION MODES 10 OUTPUT(PLOT) 11 CSCALE 2.0 12 PLOTTER NASTRAN 13 SET 1 = QUAD4 14 PTITLE = STRUCTURAL ELEMENTS 15 FIND SCALE, ORIGIN 1, SET 1 16 PLOT MODAL 0 17 OUTPUT (XYOUT) 18 CSCALE 2.0 19 PLOTTER NASTRAN 20 CURVELINESYMBOL = -6 21 YTTITLE = DAMPING G 22 YBTITLE = FREQUENCY F HZ 23 XTITLE = VELOCITY V (FT/SEC) 24 XTGRID LINES = YES 25 XBGRID LINES = YES 26 YTGRID LINES = YES 27 YBGRID LINES = YES 28 UPPER TICS = -1 29 TRIGHT TICS = -1 30 BRIGHT TICS = -1 31 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 32 BEGIN BULK EXAMPLE HA145FB: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 3 PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO WITH WKK 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ INCLUDE PLATE_STRUCT.DAT $ $ * * MASS AND INERTIA PROPERTIES * * $ $ $ $ THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES FOR LINEAR, $ $ TEMPERATURE-INDEPENDENT, ISOTROPIC MATERIALS. IT LISTS $ $ YOUNGS MODULUS, THE SHEAR MODULUS, POISSONS RATIO, $ $ MASS DENSITY, THERMAL EXPANSION COEFFICIENT AND STRUCTURAL $ $ ELEMENT DAMPING. $ $ $ $ MID E G NU RHO A TREF GE MAT1 1 9.2418+63.4993+6 0.097464 $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $ $ THE SPC1 ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS. $ $ IT LISTS THE DOF COMPONENTS TO BE CONSTRAINED AND THE GRID $ $ POINT NO. $ $ $ $ SID C G1 G2 ETC

7-78

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10

.

Flutter Analysis Sample Problems

$

SPC1 1 12345 9 SPC1 1 12345 25 SPC1 1 6 1 THRU 40 $ $ THE ASET1 ENTRY DEFINES DOFS THAT THE USER DESIRES TO BE $ INDEPENDENT. IT LISTS THE DOF COMPONENT NUMBERS AND THE $ GRID ID NO.S. $ $ V ID1 THRU ID2 ASET1 3 1 THRU 8 ASET1 3 10 THRU 24 ASET1 3 26 THRU 40 $ $ * STRUCTURAL DAMPING * $ $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUDING $ STRUCTURAL DAMPING IN THE EQUATIONS OF MOTION (SEE THE $ HANDBOOK FOR DYNAMIC ANALYSIS, SECTION 3.2.2). IF SET TO $ -1, MODAL STRUCTURAL DAMPING IS INCLUDED AS THE IMAGINARY $ PART OF A COMPLEX STIFFNESS MATRIX. $ $ N V1 V2 PARAM KDAMP +1 $ $ THE TABDMP1 ENTRY AND TABLE DEFINE MODAL DAMPING AS A $ TABULAR FUNCTION OF FREQUENCY. THE DAMPING VALUES ARE $ LINEAR BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE $ ZERO OUTSIDE THE TABULATED FREQUENCY RANGE. $ $ ID TABDMP1 2000 $ F1 G1 F2 G2 ETC ENDT +T2000 0.0 0.01 1000.0 0.01 ENDT $ * * * AERODYNAMIC DATA * * * $ $ $ (LB-IN-SEC SYSTEM) $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD $ AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=0 INDICATES $ THAT THE MODEL IS MOUNTED AS AN ISOLATED WING; SYMXY=0 $ INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH FROM THE $ FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 0 2.07055 1.145-7 1 $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET LATTICE AERODYNAMICS. $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS. $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ EXTRA POINT IDS. $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 101 1 0 10 10 1 $ ( FWD LEFT POINT ) ROOTCHORD ( FWD RIGHT POINT ) TIP CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1 $ $ * * * CORRECTION FACTORS FOR THICKNESS * * * $ DMI WTFACT 0 3 1 0 200 1 DMI WTFACT 1 1 1.2020 1.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +TDP +T2000 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +CA101

$ $ $ $

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1.0224 0.8429 1.1122 0.9327 1.2020 1.0224 0.8429 1.1122 0.9327 1.2020 1.0224 0.8429 $

0

7-80

0.9776 0.7980 1.0673 0.8878 1.1571 0.9776 0.7980 1.0673 0.8878 1.1571 0.9776 0.7980

0.9776 0.7980 1.06739 0.8878 1.1571 0.9776 0.7980 1.06739 0.8878 1.1571 0.9776 0.7980

0.9327 1.2020 1.0224 0.8429 1.1122 0.9327 1.2020 1.0224 0.8429 1.1122 0.9327

0.9327 1.2020 1.0224 0.8429 1.1122 0.9327 1.2020 1.0224 0.8429 1.1122 0.9327

0.8878 1.1571 0.9776 0.7980 1.0673 0.8878 1.1571 0.9776 0.7980 1.0673 0.8878

0.8878 1.1571 0.9776 0.7980 1.06739 0.8878 1.1571 0.9776 0.7980 1.06739 0.8878

0.8429 1.1122 0.9327 1.2020 1.0224 0.8429 1.1122 0.9327 1.2020 1.0224 0.8429

$ $ * SURFACE SPLINE FIT ON THE WING * $ $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $ GRID POINTS ON THE SETG ENTRY TO THE REGION DEFINED BY $ AERODYNAMIC BOXES 101 THRU 200 OF THE REGION ON THE $ CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE $ SPLINE IS TO BE IMPOSED. $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 100 101 101 200 100 0.0 $ $ THE SET1 ENTRY DEFINES THE SET OF STRUCTURAL GRID POINTS TO $ BE USED BY THE SURFACE SPLINE FOR INTERPOLATION. $ $ SID G1 G2 ETC SET1 100 2 4 6 8 9 11 13 +S1 15 18 20 22 24 25 27 29 +S2 31 34 36 38 40 $ * * AERODYNAMIC DATABASE * * $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ TO GENERATE GENERALIZED AERO FORCE MATRICES. $ $ M1 M2 M3 ETC MKAERO1 1.3 $ K1 K2 K3 K4 ETC +MK .03 .04 .05 $ $ * * VIBRATION ANALYSIS * * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE. IN THIS CASE THE $ MODIFIED GIVENS METHOD. THREE MODES ARE DESIRED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 NO EIGR 10 MGIV 6 $ NORM +ER MAX $ $ OPPHIPA = 1 (WHEN ACCOMPANIED BY A DISPLACEMENT REQUEST IN $ THE CASE CONTROL DECK), WILL PRODUCE THE VIBRATION MODE $ DISPLACEMENTS IN THE OUTPUT. $ PARAM OPPHIPA 1 $ $ * * FLUTTER ANALYSIS * * $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. $ PARAM LMODES 4 $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE. $ $ SID METHOD DENS MACH RFRQ IMETH NVALUE EPS FLUTTER 30 PK 1 2 3 L 3 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 .20606 FLFACT 2 1.3 FLFACT 3 14400. 15600. 16800. 18000. 19200. 20400. $ $ THE PARAMETER VREF IS USED TO SCALE THE ANALYSIS UNITS OF $ VELOCITY, HERE IN/S, TO THE UNITS DESIRED IN THE OUTPUT $ FLUTTER SUMMARY TABLES, FT/S. PARAM VREF 12.0 ENDDATA INPUT BULK DATA CARD COUNT = 341

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$ $ $ $ $ $ $ $ $ $ $ $ +S1 +S2 $ $ $ $ $ $ +MK $ $ $ $ $ $ $ $ $ +ER $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO VELOCITY $ $ $ $

Flutter Analysis Sample Problems

Listing 7-23. Input Data for the 15-Degree Sweptback Wing with ZONA51 Aerodynamics and Thickness Correction EXAMPLE HA145FB: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 11 PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO WITH WKK 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AERO 0 2.07055 1.145-7 1 2ASET1 3 1 THRU 8 3ASET1 3 10 THRU 24 4ASET1 3 26 THRU 40 5CAERO1 101 1 0 10 10 1 +CA101 6+CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 7CQUAD4 1 1 1 2 10 9 +M00000 8+M00000 0.0 0.0 .041 .041 9CQUAD4 2 1 2 3 11 10 +M00001 10+M00001 0.0 0.0 .041 .041 11CQUAD4 3 1 3 4 12 11 +M00002 12+M00002 0.0 0.0 .041 .041 13CQUAD4 4 1 4 5 13 12 +M00003 14+M00003 0.0 0.0 .041 .041 15CQUAD4 5 1 5 6 14 13 +M00004 16+M00004 0.0 0.0 .041 .041 17CQUAD4 6 1 6 7 15 14 +M00005 18+M00005 0.0 0.0 .041 .041 19CQUAD4 7 1 7 8 16 15 +M00006 20+M00006 0.0 0.0 .041 .041 21CQUAD4 8 1 9 10 18 17 22CQUAD4 9 1 10 11 19 18 23CQUAD4 10 1 11 12 20 19 24CQUAD4 11 1 12 13 21 20 25CQUAD4 12 1 13 14 22 21 26CQUAD4 13 1 14 15 23 22 27CQUAD4 14 1 15 16 24 23 28CQUAD4 15 1 17 18 26 25 29CQUAD4 16 1 18 19 27 26 30CQUAD4 17 1 19 20 28 27 31CQUAD4 18 1 20 21 29 28 32CQUAD4 19 1 21 22 30 29 33CQUAD4 20 1 22 23 31 30 34CQUAD4 21 1 23 24 32 31 35CQUAD4 22 1 25 26 34 33 +M00007 36+M00007 .041 .041 0.0 0.0 37CQUAD4 23 1 26 27 35 34 +M00008 38+M00008 .041 .041 0.0 0.0 39CQUAD4 24 1 27 28 36 35 +M00009 40+M00009 .041 .041 0.0 0.0 41CQUAD4 25 1 28 29 37 36 +M00010 42+M00010 .041 .041 0.0 0.0 43CQUAD4 26 1 29 30 38 37 +M00011 44+M00011 .041 .041 0.0 0.0 45CQUAD4 27 1 30 31 39 38 +M00012 46+M00012 .041 .041 0.0 0.0 47CQUAD4 28 1 31 32 40 39 +M00013 48+M00013 .041 .041 0.0 0.0 49DMI WTFACT 0 3 1 0 200 1 50DMI WTFACT 1 1 1.2020 1.2020 1.1571 1.1571 1.1122 +000001 51++0000011.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 +000002 52++0000020.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 +000003 53++0000031.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 +000004 54++0000041.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 +000005 55++0000050.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 +000006 56++0000061.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 +000007 57++0000070.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 +000008 58++0000081.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 +000009 59++0000091.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 +000010 60++0000100.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 +000011 61++0000111.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 +000012 62++0000120.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 +000013 63++0000131.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 +000014 64++0000141.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 +000015 65++0000150.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 +000016 66++0000161.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 +000017 67++0000170.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 +000018 68++0000181.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 +000019 69++0000191.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 +000020 70++0000200.8429 0.7980 0.7980 1.2020 1.2020 1.1571 1.1571 1.1122 +000021 71++0000211.1122 1.0673 1.06739 1.0224 1.0224 0.9776 0.9776 0.9327 +000022 72++0000220.9327 0.8878 0.8878 0.8429 0.8429 0.7980 0.7980 1.2020 +000023 73++0000231.2020 1.1571 1.1571 1.1122 1.1122 1.0673 1.06739 1.0224 +000024 74++0000241.0224 0.9776 0.9776 0.9327 0.9327 0.8878 0.8878 0.8429 +000025 75++0000250.8429 0.7980 0.7980 +000026 76EIGR 10 MGIV 6 +ER 77+ER MAX 78FLFACT 1 .20606 DENSITY 79FLFACT 2 1.3 MACH NO 80FLFACT 3 14400. 15600. 16800. 18000. 19200. 20400. VELOCITY 81FLUTTER 30 PK 1 2 3 L 3

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828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141-

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID MAT1 MKAERO1 +MK PAERO1 PARAM PARAM PARAM PARAM PARAM PARAM PSHELL SET1 +S1 +S2 SPC1 SPC1 SPC1 SPLINE1 TABDMP1 +T2000 ENDDATA TOTAL COUNT=

1 0.0 0.0 2 .211491 .7893 3 .422983 1.5786 4 .634474 2.3679 5 .845966 3.1572 6 1.05746 3.9465 7 1.26895 4.7358 8 1.48044 5.5251 9 .258819 0.0 10 .47031 .7893 11 .681802 1.5786 12 .893293 2.3679 13 1.10478 3.1572 14 1.31628 3.9465 15 1.52777 4.7358 16 1.73926 5.5251 17 1.03528 0.0 18 1.24677 .7893 19 1.45826 1.5786 20 1.66975 2.3679 21 1.88124 3.1572 22 2.09273 3.9465 23 2.30422 4.7358 24 2.51572 5.5251 25 1.81173 0.0 26 2.02322 .7893 27 2.23471 1.5786 28 2.44621 2.3679 29 2.6577 3.1572 30 2.86919 3.9465 31 3.08068 4.7358 32 3.29217 5.5251 33 2.07055 0.0 34 2.28204 .7893 35 2.49353 1.5786 36 2.70502 2.3679 37 2.91652 3.1572 38 3.12801 3.9465 39 3.3395 4.7358 40 3.55099 5.5251 1 9.2418+63.4993+6 1.3 .03 .04 .05 1 COUPMASS1 KDAMP +1 LMODES 4 OPPHIPA 1 VREF 12.0 WTMASS .0025901 1 1 .041 1 100 2 4 6 15 18 20 22 31 34 36 38 1 6 1 THRU 1 12345 9 1 12345 25 100 101 101 200 2000 0.0 0.01 1000.0 0.01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.097464 +MK

1 9 25

8 24 40 40 100

11 27

13 29

+S1 +S2

0.0 +T2000

ENDT

142

Listing 7-24. Sorted Bulk Data for the 15-Degree Sweptback Wing with ZONA51 Aerodynamics and Thickness Correction EXAMPLE HA145FB: HALF SPAN 15-DEG SWEPT UNTAPERED WING PK-METHOD OF FLUTTER ANALYSIS, ZONA51 AERO WITH WKK 0.041 IN AL PLATE W/BEVELLED LEADING AND TRAILING EDGES FLUTTER SUMMARY POINT = 1 MACH NUMBER = 1.3000 DENSITY RATIO = 2.0606E-01 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.0287 3.4859509E+01 1.2000000E+03 -2.5494257E-01 6.3504665E+01 0.0302 3.3135532E+01 1.3000000E+03 -2.8689441E-01 7.2376068E+01 0.0336 2.9803984E+01 1.4000000E+03 -3.6845586E-01 8.6656143E+01 0.0351 2.8508036E+01 1.5000000E+03 -7.7057552E-01 9.7066551E+01 0.0321 3.1130293E+01 1.6000000E+03 -1.1467444E+00 9.4816177E+01 0.0290 3.4471218E+01 1.7000000E+03 -1.4854165E+00 9.0978317E+01 POINT = 2 MACH NUMBER = 1.3000 DENSITY RATIO = 2.0606E-01 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.0729 1.3718625E+01 1.2000000E+03 -9.8030336E-02 1.6136760E+02 0.0619 1.6153862E+01 1.3000000E+03 -9.3216203E-02 1.4846107E+02 0.0502 1.9918922E+01 1.4000000E+03 -3.9160348E-02 1.2966055E+02 0.0415 2.4125406E+01 1.5000000E+03 3.1041557E-01 1.1469969E+02 0.0378 2.6469095E+01 1.6000000E+03 6.0085303E-01 1.1151326E+02 0.0348 2.8695536E+01 1.7000000E+03 8.3241701E-01 1.0928995E+02 POINT = 3 MACH NUMBER = 1.3000 DENSITY RATIO = 2.0606E-01 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.1186 8.4330111E+00 1.2000000E+03 -3.4349225E-02 2.6250903E+02 0.1098 9.1070509E+00 1.3000000E+03 -3.5831008E-02 2.6333658E+02

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51

METHOD = PK COMPLEX -5.0862518E+01 -6.5232948E+01 -1.0030780E+02 -2.3498206E+02 -3.4158511E+02 -4.2455704E+02 METHOD = PK COMPLEX -4.9696602E+01 -4.3476433E+01 -1.5951601E+01 1.1185506E+02 2.1049638E+02 2.8580582E+02 METHOD = PK COMPLEX -2.8327686E+01 -2.9642860E+01

EIGENVALUE 3.9901160E+02 4.5475229E+02 5.4447662E+02 6.0988715E+02 5.9574762E+02 5.7163367E+02 EIGENVALUE 1.0139025E+03 9.3280847E+02 8.1468127E+02 7.2067944E+02 7.0065851E+02 6.8668903E+02 EIGENVALUE 1.6493929E+03 1.6545927E+03

Flutter Analysis Sample Problems

0.1023 0.0959 0.0903 0.0854

9.7723389E+00 1.0428099E+01 1.1073617E+01 1.1708231E+01

1.4000000E+03 1.5000000E+03 1.6000000E+03 1.7000000E+03

-3.7354562E-02 -3.8922511E-02 -4.0535599E-02 -4.2193513E-02

2.6428659E+02 2.6535773E+02 2.6654843E+02 2.6785715E+02

-3.1014778E+01 -3.2447594E+01 -3.3943966E+01 -3.5505760E+01

1.6605618E+03 1.6672919E+03 1.6747732E+03 1.6829962E+03

Listing 7-25. Output for the 15-Degree Sweptback Wing with ZONA51 Aerodynamics and Thickness Correction

7.9 High Supersonic Flutter Analysis of the 15-Degree Sweptback Wing Using Piston Theory (Example HA145G) The flutter tests of Tuovila and McCarty (1955) included an untapered magnesium plate wing with 15 degrees of sweep at a Mach number of 3.0. Since this planform has already been considered in the flutter analyses of Examples HA145E and HA145F, the same structural model (via INCLUDE PLATE_STRUCT.DAT, Listing 6-11) is used here. (See Figure 6-4) However, because the material is changed from aluminum to magnesium, the density and moduli of elasticity are different. This model was also used to demonstrate the Piston Theory by Rodden, Harder, and Bellinger (1979) with a “stick” structural model. Those results will be compared to the CQUAD4 plate model used here. The PK-method of flutter analysis will be used to compare the calculated flutter speed with the test value at the same air density (altitude) and Mach number. The version of Piston Theory implemented in NX Nastran is based on a rigid chord with a rigid trailing edge flap without aerodynamic balance. The example wing has no control surface. It also is not rigid in the chordwise direction but can have a camber mode of vibration due to chordwise bending (cambering) of the plate elements. To obtain a mean chord line, a linear spline was used to interpolate between representative fore and aft grid points on the chord in the earlier solution [Example HA75G in Rodden (1987)]. It is possible to use multiple CAERO5 entries to represent a cambering surface with certain limitations, and the present example is a reconsideration of Example HA75G using four CAERO5 entries. The nominal properties of magnesium for the MAT1 entry include the moduli of elasticity, E = 6.0 × 106 and G = 2.4 × 106psi, and a density of 0.064 lb/in3. Using the grid point weight generator PARAM,GRDPNT,17 and the nominal density, the weight and the inertial properties with respect to GRID 17 are obtained and include a weight of 0.0262662 lb. The adjusted density to give the experimental weight of 0.0257 lb becomes 0.0626202 lb/in3. With the adjusted density and nominal moduli, a vibration analysis yields a second mode frequency of 211.734 Hz. The second experimental frequency is 218 Hz, so adjusting the moduli by the square of the frequency ratio gives E = 6.3604 × 106 and G = 2.5442 × 106 psi and, consequently, a model tuned to the experimental data. The calculated first and third frequencies become 35.5 and 273.7 Hz, which compare favorably with the measured values of 37 and 270 Hz. However, the three frequencies measured on the magnesium model tested at Mach 2.0 were only 35, 205, and 234 Hz; the variations in test frequencies among nominally identical models suggests a sensitivity of the installation in the wind tunnel support fixture. Two further details are necessary to include the structural damping, which was not measured but that is assumed to be g = 0.01 up to a frequency of 1000.0 rad/s. The Case Control Command SDAMP = 2000 invokes the tabulated damping TABDMP1 entry, and the damping parameter PARAM,KDAMP,+1 (the default value) selects the equivalent viscous form of the structural damping appropriate to the PK-method of flutter analysis. The AERO entry specifies the basic coordinate system as the aerodynamic coordinate system, the reference chord of 2.0706 in., and the sea-level density; no symmetry considerations are needed since the Piston Theory is a strip theory that does not have spanwise interactions. Four CAERO5 entries divide the wing into four chordwise regions and six equal width strips. The first CAERO5

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entry describes the forward beveled area of the wing, i.e., the forward one-eighth chord. The second CAERO5 entry describes the forward constant thickness area of the wing between the edge of the bevel to the midchord. The third CAERO5 entry describes the aft constant thickness area of the wing between the midchord and the edge of the aft bevel. Finally, the fourth CAERO5 entry describes the aft beveled area of the wing. The four CAERO5 entries prescribe the six equal width strips, Piston Theory with the Van Dyke correction (NTHRY = 2) and a sweep correction, and the thickness integrals on AEFACT entries; the continuation entries list the coordinates of the root and tip forward edges and the chords there for each of the chordwise regions. The thickness integrals in any analysis of a cambering surface must be input by the user; the automatic generation of the thickness integrals for a typical airfoil can no longer be used since the cambering airfoil now consists of several independent segments. The “airfoil” shape of this model consists of the beveled leading and trailing edge regions with constant thickness in between, and the thickness integrals are derived in the four chordwise regions from the general expressions:

where ξ is the dimensionless distance along the chord and gξ is the slope of the dimensionless semi-thickness distribution g(ξ). The dimensional thickness is 2τcg(ξ) where τ = t/c is the dimensionless thickness ratio. The airfoil thickness is t = 0.041 in.; its chord is c = 2.0706 in., and it is beveled through the first and last eighths of the chord. The thickness ratio is τ = 0.041/2.0706 = 0.019801.. The slope in the leading edge region is gξ = (c/2)/(1/8) = 0.079204; the slope in the trailing edge region is -0.079204, and the slope in between is zero. The first set of integrals is evaluated over the forward region and the limit ξ = 1.0 occurs at the one-eighth chord. The integrals for the forward region are:

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These are entered on the AEFACT 601 entry. Because the slope of the airfoil is zero in the two intermediate chordwise regions, all of the thickness integrals are zero, and these are entered on AEFACT 602 and 603. Because the aft slope is -0.079206 in the aft beveled region, the thickness integrals are the same as in the forward region except that the first three have the opposite sign:

These values are entered on AEFACT 604. The PAERO5 entry specifies a steady trim angle of attack of 0.0 radians in the NALPHA and LALPHA fields and on the AEFACT 701 entry. Since the thickness integrals are input, no thickness data (NXlS, LXlS, NTAUS, and LTAUS) are required. On the continuation entry, CAOCi = 0.0 for each strip since there is no control surface in any chordwise region. The aerodynamic matrix database generation is prescribed on the MKAERO1 entry for the test Mach number of 3.0 and a reduced frequency range from 0.001 to 0.10, which includes the test value of 0.0399. The surface spline is the appropriate one for a cambering surface, so four SPLINE1 entries are used, and each SPLINE1 entry corresponds to the aerodynamic elements in its corresponding CAERO5 entry. The SET1 entry for the splines lists the 20 grid points used in the Mach Box Example HA145F . For the vibration analysis, the Modified Givens method, MGlV, on the ElGR entry is chosen. Six modes are requested that have the largest component normalized to unity. The parameter PARAM,OPPHIPA,1, along with a DISP request in the Case Control Section, provides the vibration mode data at both the structural and aerodynamic grid points for printing and plotting. The flutter analysis uses the lowest five of the six vibration modes as prescribed by PARAM,LMODES,5. The FLUTTER entry specifies the PK-method for the test density on FLFACT 1, the test Mach number on FLFACT 2, and the series of velocities on FLFACT 3. It also requests that three flutter eigenvalues be output from the five mode formulation. The negative velocity on FLFACT 3 requests that the flutter mode be printed at that velocity. The parameter PARAM,VREF,12.0 specifies that the Flutter Summary velocities will be divided by

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12.0 in order to convert units to ft/s for comparison with the published test flutter speed. The last entry in the Bulk Data Section is ENDDATA.

Case Control Commands The first three commands in the Case Control Section are title commands. ECHO = BOTH requests the annotated and sorted Bulk Data be printed in the output. SPC = 1 selects the constraints on the wing root deflections and on the plate element in-plane rotations. METHOD = 10 selects the eigenvalue method for the vibration analysis. SET 10 = 1 THRU 1000 and DISP = 10, along with PARAM,OPPHlPA,1 in the Bulk Data Section, request the vibration and flutter eigenvector output. The OUTPUT(PLOT) entry identifies the plot packet for the structure plotter. The plot requests are identical to those of Example HA145E . The Executive Control Section begins with the problem identification, ID NXN, HA145G. TIME 5 limits the computing time to 5.0 CPU minutes, and SOL 145 requests the Structured Aerodynamic Flutter DMAP sequence. The CEND statement completes the Executive Control Section.

Output The input data for this example are shown in Listing 7-26, followed by the sorted Bulk Data entries in Listing 7-27 and then selected output in Listing 7-28. The output data shown are discussed below. The output begins with the weight data that were used to match the measured weight. Next are the vibration frequencies and three of the six vibration modes that were calculated, including the modes of the six aerodynamic strips (one-quarter chord deflections and rotations) in each of the four chordwise regions. The flutter solution includes the complex modal participation factors that were requested via the negative velocity at V = 25200 in/s = 2100 ft/s, the Flutter Summaries, and one of the complex flutter modes of the structural grid points. The five mode flutter solution for the first three roots is presented in three Flutter Summaries and is preceded by the complex modal participation factors at V = 2100 ft/s and followed by the complex deflections at the structural grid points, also at V = 2100 ft/s. The Flutter Summary for the second mode (Point 2) presents the critical stability data that are plotted in Figure 7-12. The calculated flutter speed is 2077 ft/s and the frequency is 148.6 Hz. The experimental flutter speed was 2030 ft/s and the frequency was 146 Hz. The results from HA75G were Vf = 1857 ft/s and ff = 145 Hz. The earlier solution of Rodden, Harder, and Bellinger (1979) using a stick model and six modes obtained a flutter speed of 2170 ft/s and frequency of 168.0 Hz. The new solution accounting for camber has shown significant improvement over the earlier solutions.

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Figure 7-12. High Speed Flutter and Damping Curves for Sweptback Wing Model by Piston Theory A difficulty that exists in NX Nastran when using multiple CAERO5 entries to represent a cambering surface is that any sweep correction (NTHRY = 2) should be based on the sweep angle of the surface leading edge, not on the sweep angle of an aft panel. This becomes a problem for a tapered wing. This example wing has an untapered planform so the four CAERO5 panels used here all have the same leading edge sweep angle, and there is no inconsistency. N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA145G $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145G $$$$$$$$ $ $ $ MODEL DESCRIPTION MODEL A OF NASA TN D-1824 $ $ HALF SPAN 15 DEGREE SWEPT WING $ $ 28 QUAD4 PANEL MODEL $ $ $ $ SOLUTION PK-FLUTTER ANALYSIS METHOD $ $ USING PISTON THEORY METHOD $ $ AERODYNAMICS AT MACH NO. 3.0 $ $ $ $ OUTPUT PLOTS OF VIBRATION MODES AND X-Y $ $ PLOTS OF V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ SOL 145 $ FLUTTER ANALYSIS CEND

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EXAMPLE HA145G: HALF SPAN 15-DEG SWEPT UNTAPERED WING PK-METHOD FLUTTER ANALYSIS, CAMBERING PISTON THEORY AERO

PAGE

3

0.041 IN MAG PLATE W/BEVELLED LEADING AND TRAILING EDGES C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145G: HALF SPAN 15-DEG SWEPT UNTAPERED WING 2 SUBT = PK-METHOD FLUTTER ANALYSIS, CAMBERING PISTON THEORY AERO 3 LABEL = 0.041 IN MAG PLATE W/BEVELLED LEADING AND TRAILING EDGES 4 ECHO = BOTH 5 SPC = 1 $ 6 METHOD = 10 $ MODIFIED GIVENS 7 FMETHOD = 30 $ PK-FLUTTER METHOD 8 SDAMP = 2000 9 SET 10 = 1 THRU 1000 10 DISP = 10 $ GRID AND AERO BOX DISPLACEMENTS IN MODES 11 $ 12 OUTPUT(XYOUT) 13 CSCALE 2.0 14 PLOTTER NASTRAN 15 CURVELINESYMBOL = 6 16 YTTITLE = DAMPING G 17 YBTITLE = FREQUENCY F HZ 18 XTITLE = VELOCITY V (FT/S) 19 XMIN = 1500. 20 XMAX = 2400. 21 YTMIN = -1.0 22 YTMAX = +1.0 23 YBMIN = 0. 24 YBMAX = 200. 25 XTGRID LINES = YES 26 XBGRID LINES = YES 27 YTGRID LINES = YES 28 YBGRID LINES = YES 29 UPPER TICS = -1 30 TRIGHT TICS = -1 31 BRIGHT TICS = -1 32 XYPLOT VG / 1(G,F) 2(G,F) 33 $ 34 BEGIN BULK EXAMPLE HA145G: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 3 PK-METHOD FLUTTER ANALYSIS, CAMBERING PISTON THEORY AERO 0.041 IN MAG PLATE W/BEVELLED LEADING AND TRAILING EDGES I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ INCLUDE PLATE_STRUCT.DAT $ $ $ THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES FOR LINEAR, $ $ TEMPERATURE-INDEPENDENT, ISOTROPIC MATERIALS. IT LISTS $ $ YOUNGS MODULUS, THE SHEAR MODULUS, POISSONS RATIO, DENSITY, $ $ THERMAL EXPANSION COEFFICIENT AND STRUCTURAL ELEMENT DAMPING. $ $ $ $ MID E G NU RHO A TREF GE MAT1 1 6.3604+62.5442+6 .0626202 MGNSIUM $ $ $ $ $ PARAMETER GRDPNT GENERATES THE INERTIAL MATRIX WITH RESPECT $ $ TO A COORDINATE SYSTEM THROUGH THE GRID POINT SPECIFIED. $ $ $ PARAM GRDPNT 17 $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $ $ THE SPC1 ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS. $ $ IT LISTS ITS ID, THE DOF COMPONENTS TO BE CONSTRAINED $ $ AND THE GRID POINT NO. $ $ $ $ SID C G1 G2 ETC SPC1 1 12345 9 SPC1 1 12345 25 SPC1 1 6 1 THRU 40 $ $ $ THE ASET1 ENTRY DEFINES DOFS THAT THE USER DESIRES TO BE $ $ IN THE ANALYSIS SET. IT LISTS THE DOF COMPONENT NUMBERS AND $ $ THE GRID ID NO.S. $ $ $ $ V ID1 THRU ID2 ASET1 3 1 THRU 8 ASET1 3 10 THRU 16 ASET1 3 18 THRU 24 ASET1 3 26 THRU 40 $ $

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$

$ $ $ $ $ $ $ $ $ PARAM $ $ $ $ $ $ $ TABDMP1 $ +T2000 $ $ $ $ $ $ $ THE AERO ENTRY SPECIFIES THE $ $ $ $ $ $ AERO $ $ $ $ $ $ $ $ $ $ $ CAERO5 $ $ +CA101 CAERO5 +CA201 CAERO5 +CA301 CAERO5 +CA401 $ $ $ AEFACT AEFACT AEFACT AEFACT $ $ $ $ $ $ $ $ PAERO5 $ +PA5 $ $ AEFACT $ $ $ $ $ $ $ MKAERO1 $ +MK $ $ $ $

*

STRUCTURAL DAMPING *

THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUDING STRUCTURAL DAMPING IN THE EQUATIONS OF MOTION (SEE THE HANDBOOK FOR DYNAMIC ANALYSIS, SECTION 3.2.2). IF SET TO -1, MODAL STRUCTURAL DAMPING IS INCLUDED AS THE IMAGINARY PART OF A COMPLEX STIFFNESS MATRIX. N KDAMP

V1 +1

V2

THE TABDMP1 ENTRY AND TABLE DEFINE MODAL DAMPING AS A TABULAR FUNCTION OF FREQUENCY. THE DAMPING VALUES ARE LINEAR BETWEEN THE FREQUENCY AND DAMPING PAIRS AND ARE EXTRAPOLATED OUTSIDE THE TABULATED FREQUENCY RANGE. ID 2000 F1 0.0

G1 0.01

F2 1000.0

G2 0.01

ETC ENDT

(LB-IN-SEC SYSTEM) * * ELEMENT GEOMETRY * * AERO COORDINATE SYSTEM, THE $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=0 INDICATES THAT THE MODEL IS MOUNTED AS AN ISOLATED WING; SYMXY=0 INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. ACSID VELOCITY REFC RHOREF SYMXZ SYMXY 0 2.0706 1.145-7 THE CAERO5 ENTRY DEFINES PANEL ELEMENTS FOR THE PISTON THEORY METHOD. IT LISTS THE ID OF THE PAERO ENTRY, THE COORDINATE SYSTEM OF POINTS 1 AND 4, THE NO. OF EQUAL STRIPS (LSPAN WOULD SPECIFY AN AEFACT ENTRY IF UNEQUAL STRIPS WERE SPECIFIED), AND AN AEFACT ENTRY WHERE THICKNESS INTEGRALS WILL BE FOUND. NTHRY SPECIFIES PISTON OR VAN DYKES THEORY WITH OR WITHOUT SWEEP EFFECTS. THE CONTINUATION ENTRY CONTAINS THE COORDINATES OF THE ROOT AND TIP LEADING EDGES AND THE CHORDS THERE. PID CP 1 0 (INBD POINT) X1 Y1 Z1 .0 .0 .0 201 1 0 .258819 .0 .0 301 1 0 1.035275.0 .0 401 1 0 1.811731.0 .0

NSPAN 6 X12 ROOTCHORD .258819 6 .776456 6 .776456 6 .258819

$ $ $ $ $ $ +TDP +T2000

ENDT

* * * AERODYNAMIC DATA * * *

EID 101

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

LSPAN

NTHRY NTHICK 2 601 +CA101 (OUTBD POINT) X43 X4 Y4 Z4 TIPCHORD 1.48044 5.52510 0.0 .258819 2 602 +CA201 1.7392595.52510 0.0 .776456 2 603 +CA301 2.5157155.52510 0.0 .776456 2 604 +CA401 3.2921715.52510 0.0 .258819 $ THE SIX THICKNESS INTEGRALS. $ $ 601 0.079204 .039602 .026401 .006273 .003137 .002091 602 0.0 -.0 -.0 .0 .0 .0 603 0.0 -.0 -.0 .0 .0 .0 604 -.079204-.039602-.026401 .006273 .003137 .002091 $ THE PAERO5 ENTRY CONTAINS THREE PAIRS OF NUMBERS SPECIFYING $ THE NUMBERS OF PARAMETERS AND THE AEFACT ENTRIES ON WHICH THEY $ ARE LISTED; THE PARAMETERS ARE LOCAL ANGLE OF ATTACK, LOCAL $ CHORD AND LOCAL THICKNESS. THE CONTINUATION ENTRY LISTS $ TRAILING EDGE CONTROL SURFACE CHORDS IN FRACTION OF LOCAL $ CHORD. $ PID NALPHA LALPHA NXIS LXIS NTAUS LTAUS 1 1 701 +PA5 CAOC1 CAOC2 CAOC3 CAOC4 CAOC5 CAOC6 0. 0. 0. 0. 0. 0. $ MACH1 ALPHA11 ALPHA12 ETC MACH2 ETC 701 3.0 0. * * FLIGHT CONDITIONS * * $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY ARE USED TO $ GENERATE GENERALIZED AERODYNAMIC FORCE MATRICES. $ $ M1 M2 M3 ETC 3.0 +MK K1 K2 K3 ETC .001 .02 .03 .04 .06 .10 $ * SURFACE SPLINE FIT ON THE WING * $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $

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$ $ $ $

$

7-90

GRID POINTS ON THE SETG ENTRY TO THE SUB-REGION DEFINED BY AERODYNAMIC BOXES 101 THRU 124 OF THE REGION ON THE CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE SPLINE IS TO BE IMPOSED. EID CAERO BOX1 BOX2 SETG DZ 100 101 101 106 100 0.0 200 201 201 206 100 0.0 300 301 301 306 100 0.0 400 401 401 406 100 0.0

$ SPLINE1 SPLINE1 SPLINE1 SPLINE1 $ THIS CORD2R ENTRY DEFINES THE SPLINE COORDINATE SYSTEM $ $ FLAGGED BY THE SPLINE2 ENTRY. LISTED ARE THE ORIGIN. A $ POINT ALONG THE Z-AXIS, AND A POINT IN THE X-Z PLANE, $ ALL IN THE RID COORDINATE SYSTEM. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 10 1.035275 0.0 0.0 1.035275 0.0 1.0 $+CRD2 C1 C2 C3 +CRD 2.07055 -.277401 1.0 $ $ THE SET1 ENTRY DEFINES THE SETS OF POINTS TO BE USED BY $ THE SPLINE FOR INTERPOLATION. $ $ SID G1 G2 ETC SET1 100 2 4 6 8 9 11 13 +S1 15 18 20 22 24 25 27 29 +S2 31 34 36 38 40 $ $ * * VIBRATION ANALYSIS * * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM; IN THIS CASE THE $ MODIFIED GIVENS METHOD. SIX MODES ARE REQUESTED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 NO EIGR 10 MGIV 6 $ NORM +ER MAX $ $ THE PARAMETER OPPHIPA,1 PRINTS THE MODAL DISPLACEMENTS $ AT THE STRUCTURAL AND AERO GRID POINTS THAT CORRESPOND $ TO THE DISP AND SET REQUESTS IN THE CASE CONTROL DECK. $ PARAM OPPHIPA 1 $ $ * * * FLUTTER ANALYSIS * * * $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. $ PARAM LMODES 5 $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE. $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 30 PK 1 2 3 L 3 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS. $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ $ SID F1 F2 F3 ETC FLFACT 1 0.391 FLFACT 2 3.0 FLFACT 3 19200. 20400. 21600. 22800. 24000. 24840. 24960. +FL3 -25200. 26400. $ $ THE PARAMETER VREF CONVERTS THE ANALYSIS UNITS OF VELOCITY $ TO THE UNITS DESIRED IN THE FLUTTER SUMMARY TABLES. $ $ N V1 V2 PARAM VREF 12.0 $ ENDDATA INPUT BULK DATA CARD COUNT = 358

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $

$ $ $ $ $ +CRD2 +CRD $ $ $ $ +S1 +S2 $ $ $ $ $ $ $ $ $ +ER $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO +FL3 $ $ $ $ $ $

Flutter Analysis Sample Problems

Listing 7-26. Input Data for the 15-Degree Sweptback Wing with Piston Theory Aerodynamics EXAMPLE HA145G: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 11 PK-METHOD FLUTTER ANALYSIS, CAMBERING PISTON THEORY AERO 0.041 IN MAG PLATE W/BEVELLED LEADING AND TRAILING EDGES S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 601 0.079204.039602 .026401 .006273 .003137 .002091 2AEFACT 602 0.0 -.0 -.0 .0 .0 .0 3AEFACT 603 0.0 -.0 -.0 .0 .0 .0 4AEFACT 604 -.079204-.039602-.026401.006273 .003137 .002091 5AEFACT 701 3.0 0. 6AERO 0 2.0706 1.145-7 7ASET1 3 1 THRU 8 8ASET1 3 10 THRU 16 9ASET1 3 18 THRU 24 10ASET1 3 26 THRU 40 11CAERO5 101 1 0 6 2 601 +CA101 12+CA101 .0 .0 .0 .258819 1.48044 5.52510 0.0 .258819 13CAERO5 201 1 0 6 2 602 +CA201 14+CA201 .258819 .0 .0 .776456 1.7392595.52510 0.0 .776456 15CAERO5 301 1 0 6 2 603 +CA301 16+CA301 1.035275.0 .0 .776456 2.5157155.52510 0.0 .776456 17CAERO5 401 1 0 6 2 604 +CA401 18+CA401 1.811731.0 .0 .258819 3.2921715.52510 0.0 .258819 19CORD2R 10 1.0352750.0 0.0 1.0352750.0 1.0 +CRD 20+CRD 2.07055 -.2774011.0 21CQUAD4 1 1 1 2 10 9 +M00000 22+M00000 0.0 0.0 .041 .041 23CQUAD4 2 1 2 3 11 10 +M00001 24+M00001 0.0 0.0 .041 .041 25CQUAD4 3 1 3 4 12 11 +M00002 26+M00002 0.0 0.0 .041 .041 27CQUAD4 4 1 4 5 13 12 +M00003 28+M00003 0.0 0.0 .041 .041 29CQUAD4 5 1 5 6 14 13 +M00004 30+M00004 0.0 0.0 .041 .041 31CQUAD4 6 1 6 7 15 14 +M00005 32+M00005 0.0 0.0 .041 .041 33CQUAD4 7 1 7 8 16 15 +M00006 34+M00006 0.0 0.0 .041 .041 35CQUAD4 8 1 9 10 18 17 36CQUAD4 9 1 10 11 19 18 37CQUAD4 10 1 11 12 20 19 38CQUAD4 11 1 12 13 21 20 39CQUAD4 12 1 13 14 22 21 40CQUAD4 13 1 14 15 23 22 41CQUAD4 14 1 15 16 24 23 42CQUAD4 15 1 17 18 26 25 43CQUAD4 16 1 18 19 27 26 44CQUAD4 17 1 19 20 28 27 45CQUAD4 18 1 20 21 29 28 46CQUAD4 19 1 21 22 30 29 47CQUAD4 20 1 22 23 31 30 48CQUAD4 21 1 23 24 32 31 49CQUAD4 22 1 25 26 34 33 +M00007 50+M00007 .041 .041 0.0 0.0 51CQUAD4 23 1 26 27 35 34 +M00008 52+M00008 .041 .041 0.0 0.0 53CQUAD4 24 1 27 28 36 35 +M00009 54+M00009 .041 .041 0.0 0.0 55CQUAD4 25 1 28 29 37 36 +M00010 56+M00010 .041 .041 0.0 0.0 57CQUAD4 26 1 29 30 38 37 +M00011 58+M00011 .041 .041 0.0 0.0 59CQUAD4 27 1 30 31 39 38 +M00012 60+M00012 .041 .041 0.0 0.0 61CQUAD4 28 1 31 32 40 39 +M00013 62+M00013 .041 .041 0.0 0.0 63EIGR 10 MGIV 6 +ER 64+ER MAX 65FLFACT 1 0.391 DENSITY 66FLFACT 2 3.0 MACH NO 67FLFACT 3 19200. 20400. 21600. 22800. 24000. 24840. 24960. +FL3 68+FL3 -25200. 26400. 69FLUTTER 30 PK 1 2 3 L 3 70GRID 1 0.0 0.0 0.0 71GRID 2 .211491 .7893 0.0 72GRID 3 .422983 1.5786 0.0 73GRID 4 .634474 2.3679 0.0 74GRID 5 .845966 3.1572 0.0 75GRID 6 1.05746 3.9465 0.0 76GRID 7 1.26895 4.7358 0.0 77GRID 8 1.48044 5.5251 0.0 78GRID 9 .258819 0.0 0.0 79GRID 10 .47031 .7893 0.0 80GRID 11 .681802 1.5786 0.0 81GRID 12 .893293 2.3679 0.0

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828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134-

Flutter Analysis Sample Problems

GRID 13 1.10478 3.1572 GRID 14 1.31628 3.9465 GRID 15 1.52777 4.7358 GRID 16 1.73926 5.5251 GRID 17 1.03528 0.0 GRID 18 1.24677 .7893 GRID 19 1.45826 1.5786 GRID 20 1.66975 2.3679 GRID 21 1.88124 3.1572 GRID 22 2.09273 3.9465 GRID 23 2.30422 4.7358 GRID 24 2.51572 5.5251 GRID 25 1.81173 0.0 GRID 26 2.02322 .7893 GRID 27 2.23471 1.5786 GRID 28 2.44621 2.3679 GRID 29 2.6577 3.1572 GRID 30 2.86919 3.9465 GRID 31 3.08068 4.7358 GRID 32 3.29217 5.5251 GRID 33 2.07055 0.0 GRID 34 2.28204 .7893 GRID 35 2.49353 1.5786 GRID 36 2.70502 2.3679 GRID 37 2.91652 3.1572 GRID 38 3.12801 3.9465 GRID 39 3.3395 4.7358 GRID 40 3.55099 5.5251 MAT1 1 6.3604+62.5442+6 MKAERO1 3.0 +MK .001 .02 .03 .04 PAERO5 1 1 701 +PA5 0. 0. 0. 0. PARAM COUPMASS1 PARAM GRDPNT 17 PARAM KDAMP +1 PARAM LMODES 5 PARAM OPPHIPA 1 PARAM VREF 12.0 PARAM WTMASS .0025901 PSHELL 1 1 .041 1 SET1 100 2 4 6 +S1 15 18 20 22 +S2 31 34 36 38 SPC1 1 6 1 THRU SPC1 1 12345 9 SPC1 1 12345 25 SPLINE1 100 101 101 106 SPLINE1 200 201 201 206 SPLINE1 300 301 301 306 SPLINE1 400 401 401 406 TABDMP1 2000 +T2000 0.0 0.01 1000.0 0.01 ENDDATA TOTAL COUNT= 135

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 .0626202

MGNSIUM +MK

.06

.10

0.

0.

+PA5

8 24 40 40 100 100 100 100

1 9 25

11 27

13 29

+S1 +S2

0.0 0.0 0.0 0.0 +T2000

ENDT

Listing 7-27. Sorted Bulk Data for the 15-Degree Sweptback Wing with Piston Theory Aerodynamics EXAMPLE HA145G: HALF SPAN 15-DEG SWEPT UNTAPERED WING PAGE 15 PK-METHOD FLUTTER ANALYSIS, CAMBERING PISTON THEORY AERO 0.041 IN MAG PLATE W/BEVELLED LEADING AND TRAILING EDGES O U T P U T F R O M G R I D P O I N T W E I G H T G E N REFERENCE POINT = 17 M O * 2.569994E-02 -6.723324E-21 0.000000E+00 0.000000E+00 0.000000E+00 -7.099739E-02 * * -6.723324E-21 2.569994E-02 0.000000E+00 0.000000E+00 0.000000E+00 1.902349E-02 * * 0.000000E+00 0.000000E+00 2.569994E-02 7.099739E-02 -1.902349E-02 0.000000E+00 * * 0.000000E+00 0.000000E+00 7.099739E-02 2.615118E-01 -7.007129E-02 0.000000E+00 * * 0.000000E+00 0.000000E+00 -1.902349E-02 -7.007129E-02 2.594858E-02 0.000000E+00 * * -7.099739E-02 1.902349E-02 0.000000E+00 0.000000E+00 0.000000E+00 2.874604E-01 * S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. X 2.569994E-02 0.000000E+00 2.762550E+00 Y 2.569994E-02 7.402154E-01 0.000000E+00 Z 2.569994E-02 7.402154E-01 2.762550E+00 I(S) * 6.537795E-02 1.751793E-02 0.000000E+00 * * 1.751793E-02 1.186710E-02 0.000000E+00 * * 0.000000E+00 0.000000E+00 7.724505E-02 * I(Q) * 7.060268E-02 * * 6.642361E-03 * * 7.724505E-02 * Q * 9.582864E-01 2.858095E-01 0.000000E+00 *

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E R A T O R

Z-C.G. 0.000000E+00 0.000000E+00 0.000000E+00

Flutter Analysis Sample Problems

* -2.858095E-01 9.582864E-01 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 1 4.966088E+04 2.228472E+02 3.546724E+01 1.597554E-05 7.933596E-01 2 2 1.876190E+06 1.369741E+03 2.180010E+02 6.074554E-06 1.139702E+01 3 3 2.956955E+06 1.719580E+03 2.736796E+02 5.465819E-06 1.616218E+01 4 4 1.694829E+07 4.116830E+03 6.552139E+02 5.314034E-06 9.006378E+01 5 5 2.949705E+07 5.431119E+03 8.643893E+02 5.243518E-06 1.546683E+02 6 6 7.178733E+07 8.472740E+03 1.348478E+03 3.104154E-06 2.228389E+02 7 7 1.101135E+08 1.049350E+04 1.670092E+03 0.0 0.0 8 8 2.209181E+08 1.486331E+04 2.365570E+03 0.0 0.0 9 9 3.014644E+08 1.736273E+04 2.763365E+03 0.0 0.0 10 10 3.820511E+08 1.954613E+04 3.110863E+03 0.0 0.0 EIGENVALUE = 4.966088E+04 CYCLES = 3.546724E+01 R E A L E I G E N V E C T O R N O . 1 POINT ID. TYPE T1 T2 T3 R1 R2 1 G 0.0 0.0 -3.177641E-03 5.710699E-02 -2.225847E-02 2 G 0.0 0.0 3.466399E-02 2.827334E-02 -1.574365E-02 3 G 0.0 0.0 1.208369E-01 1.687295E-01 -3.860027E-02 4 G 0.0 0.0 2.424595E-01 1.223329E-01 -2.303771E-02 5 G 0.0 0.0 3.901408E-01 2.282930E-01 -4.816557E-02 6 G 0.0 0.0 5.602161E-01 1.802223E-01 -3.166129E-02 7 G 0.0 0.0 7.362689E-01 2.426241E-01 -4.697696E-02 8 G 0.0 0.0 9.192188E-01 1.988754E-01 -3.358189E-02 9 G 0.0 0.0 0.0 0.0 0.0 10 G 0.0 0.0 4.019029E-02 9.116539E-02 -3.023629E-02 11 G 0.0 0.0 1.287552E-01 1.183675E-01 -1.831846E-02 12 G 0.0 0.0 2.497337E-01 1.714814E-01 -3.489073E-02 13 G 0.0 0.0 4.012302E-01 1.916025E-01 -3.536316E-02 14 G 0.0 0.0 5.693870E-01 2.126201E-01 -3.980973E-02 15 G 0.0 0.0 7.475079E-01 2.164140E-01 -3.917040E-02 16 G 0.0 0.0 9.288043E-01 2.211759E-01 -4.038252E-02 17 G 0.0 0.0 -3.139184E-04 6.098017E-02 -6.990873E-04 18 G 0.0 0.0 5.307267E-02 7.231662E-02 -2.554121E-03 19 G 0.0 0.0 1.423508E-01 1.461928E-01 -1.677333E-02 20 G 0.0 0.0 2.734013E-01 1.725224E-01 -2.565950E-02 21 G 0.0 0.0 4.279933E-01 2.013863E-01 -3.327638E-02 22 G 0.0 0.0 5.991349E-01 2.120074E-01 -3.647467E-02 23 G 0.0 0.0 7.779750E-01 2.198114E-01 -3.903528E-02 24 G 0.0 0.0 9.598176E-01 2.195428E-01 -3.923723E-02 25 G 0.0 0.0 0.0 0.0 0.0 26 G 0.0 0.0 4.735721E-02 1.203464E-01 1.591588E-02 27 G 0.0 0.0 1.539442E-01 1.485789E-01 -1.306415E-02 28 G 0.0 0.0 2.916331E-01 1.886990E-01 -2.129181E-02 29 G 0.0 0.0 4.529409E-01 2.041422E-01 -3.071789E-02 30 G 0.0 0.0 6.273171E-01 2.181408E-01 -3.591881E-02 31 G 0.0 0.0 8.079913E-01 2.187324E-01 -3.807590E-02 32 G 0.0 0.0 9.904229E-01 2.223233E-01 -3.940554E-02 33 G 0.0 0.0 -3.292926E-03 3.037128E-02 1.764832E-02 34 G 0.0 0.0 4.410950E-02 1.026832E-01 1.694509E-02 0.0 35 G 0.0 0.0 1.569097E-01 1.761944E-01 -1.333694E-02 36 G 0.0 0.0 2.970206E-01 1.706534E-01 -1.712837E-02 37 G 0.0 0.0 4.611472E-01 2.267395E-01 -3.401155E-02 38 G 0.0 0.0 6.362092E-01 1.984645E-01 -3.138098E-02 39 G 0.0 0.0 8.184620E-01 2.411798E-01 -4.306755E-02 40 G 0.0 0.0 1.000000E+00 1.976597E-01 -3.439136E-02 101 G 0.0 0.0 1.783676E-02 0.0 -1.343707E-02 102 G 0.0 0.0 9.879901E-02 0.0 -1.880716E-02 103 G 0.0 0.0 2.328612E-01 0.0 -2.450174E-02 104 G 0.0 0.0 4.083946E-01 0.0 -3.175720E-02 105 G 0.0 0.0 6.061424E-01 0.0 -3.663930E-02 106 G 0.0 0.0 8.159464E-01 0.0 -4.059823E-02 201 G 0.0 0.0 2.388476E-02 0.0 -1.547890E-02 202 G 0.0 0.0 1.066445E-01 0.0 -1.923210E-02 203 G 0.0 0.0 2.439832E-01 0.0 -3.079459E-02 204 G 0.0 0.0 4.213656E-01 0.0 -3.445108E-02 205 G 0.0 0.0 6.208205E-01 0.0 -3.873397E-02 206 G 0.0 0.0 8.315884E-01 0.0 -3.993377E-02 301 G 0.0 0.0 3.023514E-02 0.0 1.538259E-03 302 G 0.0 0.0 1.174660E-01 0.0 -8.564179E-03 303 G 0.0 0.0 2.661315E-01 0.0 -2.238087E-02 304 G 0.0 0.0 4.478103E-01 0.0 -3.307833E-02 305 G 0.0 0.0 6.505368E-01 0.0 -3.650501E-02 306 G 0.0 0.0 8.620739E-01 0.0 -3.886171E-02 401 G 0.0 0.0 2.255161E-02 0.0 1.874720E-02 402 G 0.0 0.0 1.231855E-01 0.0 -1.207933E-02 403 G 0.0 0.0 2.796980E-01 0.0 -2.235178E-02 404 G 0.0 0.0 4.692842E-01 0.0 -3.322077E-02 405 G 0.0 0.0 6.739140E-01 0.0 -3.669397E-02 406 G 0.0 0.0 8.872889E-01 0.0 -3.880873E-02 EIGENVALUE = 1.876190E+06 CYCLES = 2.180010E+02 R E A L E I G E N V E C T O R N O . 2 POINT ID. TYPE T1 T2 T3 R1 R2 1 G 0.0 0.0 1.076519E-02 -2.433207E-01 8.325648E-02 2 G 0.0 0.0 -1.455797E-01 -9.741305E-02 -1.021172E-02 3 G 0.0 0.0 -4.042479E-01 -5.123186E-01 -4.539555E-02 4 G 0.0 0.0 -6.051180E-01 -5.583264E-02 -2.652566E-01 5 G 0.0 0.0 -6.545121E-01 -1.959130E-01 -2.932936E-01 6 G 0.0 0.0 -5.606913E-01 2.297285E-01 -4.505970E-01

NX Nastran Aeroelastic Analysis User’s Guide

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R3 0.0 0.0 0.0 0.0 0.0 0.0

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Flutter Analysis Sample Problems

7 G 0.0 8 G 0.0 9 G 0.0 10 G 0.0 11 G 0.0 12 G 0.0 13 G 0.0 14 G 0.0 15 G 0.0 16 G 0.0 17 G 0.0 18 G 0.0 19 G 0.0 20 G 0.0 21 G 0.0 22 G 0.0 23 G 0.0 24 G 0.0 25 G 0.0 26 G 0.0 27 G 0.0 28 G 0.0 29 G 0.0 30 G 0.0 31 G 0.0 32 G 0.0 33 G 0.0 34 G 0.0 35 G 0.0 36 G 0.0 37 G 0.0 38 G 0.0 39 G 0.0 40 G 0.0 101 G 0.0 102 G 0.0 103 G 0.0 104 G 0.0 105 G 0.0 106 G 0.0 201 G 0.0 202 G 0.0 203 G 0.0 204 G 0.0 205 G 0.0 206 G 0.0 301 G 0.0 302 G 0.0 303 G 0.0 304 G 0.0 305 G 0.0 306 G 0.0 401 G 0.0 402 G 0.0 403 G 0.0 404 G 0.0 405 G 0.0 406 G 0.0 EIGENVALUE = 2.956955E+06 CYCLES = 2.736796E+02 POINT ID. TYPE T1 1 G 0.0 2 G 0.0 3 G 0.0 4 G 0.0 5 G 0.0 6 G 0.0 7 G 0.0 8 G 0.0 9 G 0.0 10 G 0.0 11 G 0.0 12 G 0.0 13 G 0.0 14 G 0.0 15 G 0.0 16 G 0.0 17 G 0.0 18 G 0.0 19 G 0.0 20 G 0.0 21 G 0.0 22 G 0.0 23 G 0.0 24 G 0.0 25 G 0.0 26 G 0.0 27 G 0.0 28 G 0.0 29 G 0.0 30 G 0.0

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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R E A L T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

-3.175942E-01 -1.491527E-02 0.0 -1.483781E-01 -3.850683E-01 -5.425583E-01 -5.742006E-01 -4.483187E-01 -2.003406E-01 1.140091E-01 -9.321876E-04 -1.469279E-01 -2.871523E-01 -3.702640E-01 -3.166552E-01 -1.281732E-01 1.599456E-01 4.924163E-01 0.0 -7.832745E-02 -1.809497E-01 -1.867020E-01 -6.006376E-02 1.926582E-01 5.203841E-01 8.721126E-01 9.889715E-03 -5.051656E-02 -1.437313E-01 -1.249291E-01 2.522993E-02 3.011479E-01 6.403315E-01 1.000000E+00 -7.455474E-02 -3.329415E-01 -5.840117E-01 -6.246455E-01 -4.857388E-01 -1.622539E-01 -6.739552E-02 -3.139974E-01 -5.065786E-01 -5.109797E-01 -3.235691E-01 1.586512E-02 -8.202633E-02 0.0 -2.446632E-01 -3.266103E-01 -2.480691E-01 2.038978E-02 3.916357E-01 -4.396769E-02 -1.499702E-01 -1.826435E-01 -1.877663E-02 2.958279E-01 7.044181E-01 E I G E N V E C T O T3 1.848864E-03 2.956194E-02 8.557376E-02 1.687198E-01 2.985356E-01 5.053338E-01 7.534627E-01 1.000000E+00 0.0 1.174052E-02 4.094337E-02 8.624984E-02 1.894967E-01 3.714224E-01 6.004590E-01 8.368752E-01 -3.193164E-04 -5.006811E-02 -1.350967E-01 -2.014132E-01 -1.781411E-01 -6.092090E-02 1.242390E-01 3.359585E-01 0.0 -1.425986E-01 -3.830405E-01 -5.516362E-01 -6.036602E-01 -5.328335E-01

1.411230E-01 -4.398463E-01 3.639295E-01 -5.100251E-01 0.0 0.0 -3.298120E-01 4.441432E-02 -2.655696E-01 -1.228902E-01 -2.060066E-01 -2.132788E-01 -1.660852E-02 -3.345524E-01 1.292873E-01 -4.122958E-01 2.525528E-01 -4.655613E-01 2.822633E-01 -4.841054E-01 -1.995024E-01 -5.667394E-03 -1.640017E-01 -5.789461E-02 -2.196950E-01 -1.335604E-01 -7.673440E-02 -2.329681E-01 6.370363E-02 -3.284229E-01 2.074918E-01 -4.103427E-01 2.773446E-01 -4.594568E-01 3.044380E-01 -4.869040E-01 0.0 0.0 -2.032576E-01 -1.179812E-01 -1.080278E-01 -1.410731E-01 1.950497E-03 -2.388912E-01 1.645762E-01 -3.305187E-01 2.657362E-01 -4.128444E-01 3.153380E-01 -4.654136E-01 3.133633E-01 -4.870962E-01 -5.189328E-02 -5.428960E-02 -1.437815E-01 -1.161583E-01 -1.364077E-01 -1.389733E-01 7.692270E-02 -2.439464E-01 1.456973E-01 -3.216818E-01 3.298752E-01 -4.239694E-01 2.736300E-01 -4.567904E-01 3.707772E-01 -4.984170E-01 0.0 -6.397898E-02 0.0 -4.646284E-02 0.0 -1.627028E-01 0.0 -2.632753E-01 0.0 -3.827342E-01 0.0 -4.416946E-01 0.0 1.568056E-02 0.0 -5.877433E-02 0.0 -2.246159E-01 0.0 -3.176940E-01 0.0 -4.413960E-01 0.0 -4.726967E-01 -8.588545E-03 0.0 0.0 -1.200168E-01 0.0 -2.111200E-01 0.0 -3.452822E-01 0.0 -4.266148E-01 0.0 -4.867139E-01 0.0 -1.116502E-01 0.0 -1.774662E-01 0.0 -2.506159E-01 0.0 -3.722009E-01 0.0 -4.317957E-01 0.0 -4.759348E-01 R N O . R1 2.670977E-02 5.589033E-02 1.325503E-01 1.826797E-01 3.099898E-01 4.263177E-01 4.660743E-01 4.713722E-01 0.0 5.608930E-02 9.363273E-02 1.623567E-01 2.984429E-01 4.069353E-01 4.597443E-01 4.626641E-01 -3.943646E-02 -4.277097E-02 -5.758484E-02 8.103919E-02 2.241181E-01 3.570502E-01 4.246789E-01 4.491850E-01 0.0 -2.840995E-01 -1.612982E-01 -1.735601E-02 1.796153E-01 3.202280E-01

3 R2 -2.156844E-03 5.979180E-02 1.615312E-01 3.067155E-01 4.122333E-01 5.094689E-01 5.890397E-01 6.281542E-01 0.0 6.934599E-02 1.812581E-01 3.327654E-01 4.335784E-01 5.297652E-01 5.964572E-01 6.355969E-01 8.194263E-03 9.249041E-02 2.725917E-01 4.064467E-01 5.116053E-01 5.819532E-01 6.283526E-01 6.523202E-01 0.0 1.535193E-01 3.663884E-01 4.943498E-01 5.819765E-01 6.310563E-01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

31 G 0.0 0.0 -3.744593E-01 4.156158E-01 6.532360E-01 0.0 32 G 0.0 0.0 -1.771789E-01 4.325322E-01 6.660703E-01 0.0 33 G 0.0 0.0 -5.231076E-03 -1.271440E-01 1.287095E-02 0.0 34 G 0.0 0.0 -1.895507E-01 -2.453643E-01 1.696592E-01 0.0 35 G 0.0 0.0 -4.825777E-01 -2.661881E-01 4.065451E-01 0.0 36 G 0.0 0.0 -6.835402E-01 3.975718E-02 5.102635E-01 0.0 37 G 0.0 0.0 -7.583392E-01 1.066423E-01 6.161198E-01 0.0 38 G 0.0 0.0 -6.965958E-01 3.873594E-01 6.297446E-01 0.0 39 G 0.0 0.0 -5.463345E-01 3.421911E-01 6.755947E-01 0.0 40 G 0.0 0.0 -3.476769E-01 5.126567E-01 6.508620E-01 0.0 101 G 0.0 0.0 1.408943E-02 0.0 1.305940E-01 0.0 102 G 0.0 0.0 6.573562E-02 0.0 1.904218E-01 0.0 103 G 0.0 0.0 1.402939E-01 0.0 3.222590E-01 0.0 104 G 0.0 0.0 2.880635E-01 0.0 4.349499E-01 0.0 105 G 0.0 0.0 5.289505E-01 0.0 5.078991E-01 0.0 106 G 0.0 0.0 8.184452E-01 0.0 5.430025E-01 0.0 201 G 0.0 0.0 -1.662521E-02 0.0 3.647701E-02 0.0 202 G 0.0 0.0 -4.449691E-03 0.0 1.785130E-01 0.0 203 G 0.0 0.0 1.400623E-02 0.0 3.347271E-01 0.0 204 G 0.0 0.0 1.136803E-01 0.0 4.657716E-01 0.0 205 G 0.0 0.0 3.221495E-01 0.0 5.550641E-01 0.0 206 G 0.0 0.0 5.934925E-01 0.0 6.124577E-01 0.0 301 G 0.0 0.0 -2.174967E-02 0.0 7.119847E-03 0.0 302 G 0.0 0.0 -1.561035E-01 0.0 2.308648E-01 0.0 303 G 0.0 0.0 -2.820587E-01 0.0 4.442662E-01 0.0 304 G 0.0 0.0 -2.821251E-01 0.0 5.473077E-01 0.0 305 G 0.0 0.0 -1.426077E-01 0.0 6.385375E-01 0.0 306 G 0.0 0.0 8.810546E-02 0.0 6.702116E-01 0.0 401 G 0.0 0.0 -6.992742E-02 0.0 1.736740E-01 0.0 402 G 0.0 0.0 -3.394837E-01 0.0 3.304215E-01 0.0 403 G 0.0 0.0 -5.833032E-01 0.0 4.665709E-01 0.0 404 G 0.0 0.0 -6.367502E-01 0.0 5.304286E-01 0.0 405 G 0.0 0.0 -5.511969E-01 0.0 5.970046E-01 0.0 406 G 0.0 0.0 -3.347389E-01 0.0 6.219214E-01 0.0 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = -1.94619E+02 8.01764E+02 EIGENVECTOR 1.00000E+00 0.00000E+00 1.78271E-01 8.55037E-02 -1.33187E-04 -5.57844E-03 5.87747E-03 4.75562E-03 6.01815E-04 -3.17303E-04 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = 2.46377E+01 9.20789E+02 EIGENVECTOR 1.00000E+00 0.00000E+00 2.74293E-01 -8.44493E-02 -2.05499E-03 -3.81114E-03 1.00665E-02 -1.16588E-03 2.97790E-04 9.50066E-06 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = -8.01337E+01 1.85544E+03 EIGENVECTOR -5.18031E-01 7.81684E-03 6.43021E-01 1.35772E-03 1.00000E+00 0.00000E+00 -2.79864E-03 9.72150E-04 3.85070E-03 -4.21105E-04 FLUTTER SUMMARY POINT = 1 MACH NUMBER = 3.0000 DENSITY RATIO = 3.9100E-01 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0259 3.8567291E+01 1.6000000E+03 -2.7556443E-01 7.6530746E+01 -6.6253532E+01 4.8085690E+02 0.0265 3.7780865E+01 1.7000000E+03 -2.8125551E-01 8.3006508E+01 -7.3343742E+01 5.2154529E+02 0.0273 3.6643333E+01 1.8000000E+03 -2.8784594E-01 9.0617615E+01 -8.1945038E+01 5.6936731E+02 0.0285 3.5062897E+01 1.9000000E+03 -2.9809615E-01 9.9963387E+01 -9.3615379E+01 6.2808850E+02 0.0305 3.2786278E+01 2.0000000E+03 -3.2533023E-01 1.1253123E+02 -1.1501311E+02 7.0705457E+02 0.0325 3.0757404E+01 2.0700000E+03 -4.1089547E-01 1.2415260E+02 -1.6026440E+02 7.8007385E+02 0.0327 3.0567787E+01 2.0800000E+03 -4.3457267E-01 1.2552623E+02 -1.7137473E+02 7.8870459E+02 0.0329 3.0359016E+01 2.1000000E+03 -4.8547682E-01 1.2760472E+02 -1.9461896E+02 8.0176416E+02 0.0324 3.0817970E+01 2.2000000E+03 -7.0313680E-01 1.3169031E+02 -2.9089987E+02 8.2743463E+02 POINT = 2 MACH NUMBER = 3.0000 DENSITY RATIO = 3.9100E-01 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0641 1.5605687E+01 1.6000000E+03 -1.0428768E-01 1.8913515E+02 -6.1966240E+01 1.1883712E+03 0.0587 1.7030817E+01 1.7000000E+03 -1.0906566E-01 1.8414018E+02 -6.3093773E+01 1.1569869E+03 0.0536 1.8643053E+01 1.8000000E+03 -1.1219984E-01 1.7811095E+02 -6.2781651E+01 1.1191041E+03 0.0486 2.0562622E+01 1.9000000E+03 -1.1105067E-01 1.7045518E+02 -5.9467724E+01 1.0710016E+03 0.0433 2.3103600E+01 2.0000000E+03 -9.2665754E-02 1.5969286E+02 -4.6489475E+01 1.0033798E+03 0.0391 2.5558243E+01 2.0700000E+03 -1.5269771E-02 1.4940823E+02 -7.1673226E+00 9.3875964E+02 0.0386 2.5885735E+01 2.0800000E+03 6.6434545E-03 1.4823065E+02 3.0937262E+00 9.3136066E+02 0.0378 2.6434685E+01 2.1000000E+03 5.3514384E-02 1.4654813E+02 2.4637730E+01 9.2078906E+02 0.0356 2.8081648E+01 2.2000000E+03 2.4752708E-01 1.4452242E+02 1.1238486E+02 9.0806116E+02 FLUTTER SUMMARY POINT = 3 MACH NUMBER = 3.0000 DENSITY RATIO = 3.9100E-01 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0971 1.0303611E+01 1.6000000E+03 -7.2536744E-02 2.8646112E+02 -6.5279022E+01 1.7998884E+03 0.0919 1.0887277E+01 1.7000000E+03 -7.5485826E-02 2.8804794E+02 -6.8309341E+01 1.8098588E+03 0.0873 1.1460998E+01 1.8000000E+03 -7.8338474E-02 2.8972449E+02 -7.1303391E+01 1.8203927E+03 0.0832 1.2024382E+01 1.9000000E+03 -8.1100658E-02 2.9149155E+02 -7.4267746E+01 1.8314955E+03 0.0795 1.2577048E+01 2.0000000E+03 -8.3778299E-02 2.9335022E+02 -7.7208984E+01 1.8431738E+03 0.0772 1.2957337E+01 2.0700000E+03 -8.5605472E-02 2.9470651E+02 -7.9257645E+01 1.8516957E+03 0.0769 1.3011213E+01 2.0800000E+03 -8.5863456E-02 2.9490402E+02 -7.9549774E+01 1.8529366E+03

NX Nastran Aeroelastic Analysis User’s Guide

7-95

Flutter Analysis Sample Problems

Chapter 7

0.0762 1.3118624E+01 2.1000000E+03 -8.6377211E-02 0.0733 1.3648745E+01 2.2000000E+03 -8.8902958E-02 COMPLEX EIGENVALUE = -1.946190E+02, 8.017642E+02 C O M P L E X POINT ID.

TYPE 1

G

2

G

3

G

4

G

5

G

6

G

7

G

8

G

9

G

10

G

11

G

12

G

13

G

14

G

15

G

16

G

17

G

18

G

19

G

20

G

21

G

22

G

23

G

24

G

25

G

26

G

27

G

28

G

29

G

30

G

31

G

32

G

33

G

34

G

35

G

36

G

37

G

38

G

39

G

40

G

COMPLEX EIGENVALUE =

7-96

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.463773E+01,

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 9.207891E+02 C O M P L E X

NX Nastran Aeroelastic Analysis User’s Guide

2.9530185E+02 2.9734808E+02

-8.0133705E+01 -8.3048393E+01

E I G E N V E C T O R (REAL/IMAGINARY) T3 -1.410351E-03 7.894858E-04 1.063089E-02 -1.094815E-02 5.268578E-02 -3.154499E-02 1.376640E-01 -4.966505E-02 2.733906E-01 -5.715957E-02 4.579122E-01 -5.239669E-02 6.774721E-01 -3.334763E-02 9.170218E-01 -7.176382E-03 0.0 0.0 1.564192E-02 -1.114328E-02 6.368729E-02 -3.002717E-02 1.554258E-01 -4.455397E-02 2.985154E-01 -5.005112E-02 4.871358E-01 -4.209771E-02 7.101295E-01 -2.202267E-02 9.501925E-01 5.340298E-03 -4.317561E-04 -4.229861E-05 2.874272E-02 -1.085238E-02 9.352801E-02 -2.203108E-02 2.084136E-01 -2.981343E-02 3.702566E-01 -2.707662E-02 5.741917E-01 -1.221086E-02 8.061138E-01 1.266234E-02 1.050619E+00 4.243905E-02 0.0 0.0 3.450759E-02 -5.306293E-03 1.231633E-01 -1.277251E-02 2.580873E-01 -1.363294E-02 4.400942E-01 -3.651563E-03 6.597461E-01 1.809240E-02 9.016489E-01 4.760719E-02 1.150860E+00 7.980063E-02 -1.594711E-03 8.014027E-04 3.601863E-02 -2.946873E-03 1.325263E-01 -9.411806E-03 2.740625E-01 -8.091401E-03 4.632013E-01 4.227079E-03 6.880094E-01 2.838272E-02 9.339656E-01 5.932839E-02 1.183870E+00 9.236998E-02

NO.

R1 1.735694E-02 -1.787749E-02 1.142149E-02 -8.048949E-03 8.154002E-02 -4.072646E-02 1.076658E-01 -9.388776E-03 1.918517E-01 -1.964901E-02 2.177706E-01 1.407673E-02 2.708908E-01 1.136816E-02 2.658870E-01 2.959932E-02 0.0 0.0 3.618494E-02 -2.517238E-02 7.146912E-02 -2.255475E-02 1.326673E-01 -1.995369E-02 1.852203E-01 -5.807779E-03 2.346580E-01 7.622537E-03 2.633609E-01 2.003030E-02 2.750471E-01 2.385877E-02 2.821299E-02 -1.462158E-02 4.436449E-02 -1.275288E-02 1.072333E-01 -1.823393E-02 1.563183E-01 -8.740323E-03 2.107036E-01 2.790774E-03 2.494487E-01 1.608346E-02 2.723979E-01 2.356734E-02 2.778838E-01 2.640017E-02 0.0 0.0 8.638198E-02 -1.450759E-02 1.282250E-01 -9.169524E-03 1.867248E-01 -1.132073E-03 2.323352E-01 1.272502E-02 2.676705E-01 2.291513E-02 2.790826E-01 2.784863E-02 2.829619E-01 2.791420E-02 2.214591E-02 -3.221692E-03 7.825027E-02 -1.036969E-02 1.513983E-01 -1.067935E-02 1.811309E-01 4.487140E-03 2.527683E-01 1.240623E-02 2.590922E-01 2.797375E-02 2.953009E-01 2.550900E-02 2.676327E-01 3.186627E-02

E I G E N V E C T O R (REAL/IMAGINARY)

NO.

1.8554363E+03 1.8682931E+03 1

R2 -8.574025E-03 6.162175E-03 -1.713100E-02 -7.133394E-04 -4.574468E-02 -3.661815E-03 -6.736546E-02 -2.137508E-02 -9.964536E-02 -2.617351E-02 -1.118616E-01 -4.095432E-02 -1.276008E-01 -4.277973E-02 -1.265991E-01 -4.916488E-02 0.0 0.0 -2.280243E-02 3.168222E-03 -3.837412E-02 -9.612271E-03 -7.097928E-02 -1.791500E-02 -9.365232E-02 -2.941626E-02 -1.136638E-01 -3.826180E-02 -1.238535E-01 -4.464025E-02 -1.291921E-01 -4.737180E-02 -1.653874E-03 -4.735950E-04 -1.212907E-02 -4.644651E-03 -3.928709E-02 -1.133757E-02 -6.554675E-02 -2.028175E-02 -9.080179E-02 -2.975247E-02 -1.098735E-01 -3.853690E-02 -1.225270E-01 -4.439050E-02 -1.284818E-01 -4.778037E-02 0.0 0.0 -4.037362E-03 -9.684628E-03 -3.727977E-02 -1.266351E-02 -6.230453E-02 -2.138029E-02 -8.851524E-02 -3.044416E-02 -1.097266E-01 -3.922326E-02 -1.226674E-01 -4.524832E-02 -1.287299E-01 -4.800774E-02 8.471685E-03 -4.244356E-03 -2.864742E-03 -9.723596E-03 -3.737863E-02 -1.280036E-02 -5.890267E-02 -2.183810E-02 -9.028995E-02 -2.995990E-02 -1.069388E-01 -4.008766E-02 -1.263746E-01 -4.484631E-02 -1.255292E-01 -4.877509E-02 2

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

POINT ID.

T1 0.0 0.0 2 G 0.0 0.0 3 G 0.0 0.0 4 G 0.0 0.0 5 G 0.0 0.0 6 G 0.0 0.0 7 G 0.0 0.0 8 G 0.0 0.0 9 G 0.0 0.0 10 G 0.0 0.0 11 G 0.0 0.0 12 G 0.0 0.0 13 G 0.0 0.0 14 G 0.0 0.0 15 G 0.0 0.0 16 G 0.0 0.0 17 G 0.0 0.0 18 G 0.0 0.0 19 G 0.0 0.0 20 G 0.0 0.0 21 G 0.0 0.0 22 G 0.0 0.0 23 G 0.0 0.0 24 G 0.0 0.0 25 G 0.0 0.0 26 G 0.0 0.0 27 G 0.0 0.0 28 G 0.0 0.0 29 G 0.0 0.0 30 G 0.0 0.0 31 G 0.0 0.0 32 G 0.0 0.0 33 G 0.0 0.0 34 G 0.0 0.0 35 G 0.0 0.0 36 G 0.0 0.0 37 G 0.0 0.0 38 G 0.0 0.0 39 G 0.0 0.0 40 G 0.0 0.0 COMPLEX EIGENVALUE = -8.013371E+01,

POINT ID.

TYPE 1

G

TYPE 1

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2

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3

G

T1 0.0 0.0 0.0 0.0 0.0

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.855436E+03 C O M P L E X T2 0.0 0.0 0.0 0.0 0.0

T3 -4.864316E-04 -8.864090E-04 -1.935482E-03 1.178259E-02 1.679465E-02 3.298222E-02 8.190697E-02 4.976239E-02 2.103971E-01 5.406227E-02 4.016670E-01 4.584313E-02 6.437665E-01 2.441062E-02 9.133591E-01 -2.540748E-03 0.0 0.0 2.791183E-03 1.209698E-02 2.939385E-02 3.161556E-02 1.052149E-01 4.495187E-02 2.431163E-01 4.777471E-02 4.418868E-01 3.687257E-02 6.883711E-01 1.498629E-02 9.597401E-01 -1.293993E-02 -4.893776E-04 7.091220E-05 1.598908E-02 1.224190E-02 6.776774E-02 2.431994E-02 1.738805E-01 3.185228E-02 3.393183E-01 2.766973E-02 5.606028E-01 1.145574E-02 8.209502E-01 -1.390482E-02 1.099195E+00 -4.342917E-02 0.0 0.0 2.777615E-02 6.987514E-03 1.070018E-01 1.655276E-02 2.405479E-01 1.801192E-02 4.338252E-01 7.827271E-03 6.780856E-01 -1.388581E-02 9.532492E-01 -4.274060E-02 1.238688E+00 -7.399355E-02 -7.002870E-04 -7.989816E-04 3.179739E-02 4.876892E-03 1.197925E-01 1.386667E-02 2.622851E-01 1.340461E-02 4.651751E-01 1.280528E-03 7.171724E-01 -2.244029E-02 9.978645E-01 -5.232643E-02 1.284913E+00 -8.429529E-02

R1 -3.344592E-03 1.970408E-02 2.486905E-03 7.882766E-03 3.546819E-02 4.186218E-02 9.879222E-02 4.917210E-03 1.714154E-01 1.565185E-02 2.361879E-01 -2.028428E-02 2.852008E-01 -1.421658E-02 3.008853E-01 -3.285909E-02 0.0 0.0 7.357983E-03 2.684067E-02 4.639218E-02 2.193078E-02 1.113185E-01 1.715015E-02 1.806540E-01 9.338284E-04 2.452551E-01 -1.221723E-02 2.875911E-01 -2.337684E-02 3.032732E-01 -2.621789E-02 1.108328E-02 1.645105E-02 2.960715E-02 1.375972E-02 8.645266E-02 1.872217E-02 1.472823E-01 6.620797E-03 2.151582E-01 -5.870537E-03 2.689784E-01 -1.897111E-02 3.001595E-01 -2.561611E-02 3.087721E-01 -2.816590E-02 0.0 0.0 6.852836E-02 1.788811E-02 1.174307E-01 9.947775E-03 1.857308E-01 2.847659E-04 2.475064E-01 -1.445053E-02 2.943220E-01 -2.413009E-02 3.113805E-01 -2.901545E-02 3.153214E-01 -2.900999E-02 1.784369E-02 4.716509E-03 6.539295E-02 1.290823E-02 1.383678E-01 1.264960E-02 1.868178E-01 -6.126273E-03 2.670472E-01 -1.280233E-02 2.917350E-01 -2.977178E-02 3.244500E-01 -2.542652E-02 3.049084E-01 -3.402103E-02

E I G E N V E C T O R (REAL/IMAGINARY) T3 1.047920E-02 -3.401038E-05 -8.347771E-02 4.614047E-04 -2.404819E-01

NO.

R1 -1.617767E-01 8.064013E-04 -2.251795E-02 2.850964E-04 -2.885968E-01

R2 -1.429546E-03 -6.787959E-03 -1.778774E-02 5.251382E-04 -4.941662E-02 2.971778E-03 -9.075058E-02 2.054230E-02 -1.275245E-01 2.294533E-02 -1.557344E-01 3.603094E-02 -1.726335E-01 3.534853E-02 -1.786004E-01 4.114426E-02 0.0 0.0 -1.888024E-02 -3.943572E-03 -4.883154E-02 9.252791E-03 -9.016272E-02 1.625828E-02 -1.251795E-01 2.624581E-02 -1.543135E-01 3.282717E-02 -1.710367E-01 3.739491E-02 -1.790564E-01 3.901645E-02 -2.163543E-03 4.344013E-04 -1.713602E-02 4.352926E-03 -5.120775E-02 9.891785E-03 -8.700360E-02 1.770299E-02 -1.222699E-01 2.552080E-02 -1.505764E-01 3.247969E-02 -1.692739E-01 3.676556E-02 -1.787179E-01 3.919668E-02 0.0 0.0 -1.454044E-02 9.100946E-03 -5.023957E-02 1.021850E-02 -8.459035E-02 1.788846E-02 -1.204318E-01 2.544118E-02 -1.509461E-01 3.253265E-02 -1.702243E-01 3.720975E-02 -1.791321E-01 3.918502E-02 3.649303E-03 4.425867E-03 -1.331488E-02 8.914609E-03 -5.029335E-02 9.916334E-03 -8.165038E-02 1.823485E-02 -1.214961E-01 2.458131E-02 -1.491573E-01 3.344839E-02 -1.733323E-01 3.644723E-02 -1.768887E-01 4.016655E-02

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

3 R2 6.362339E-02 -2.729512E-04 6.046910E-02 2.436861E-05 1.501465E-01

NX Nastran Aeroelastic Analysis User’s Guide

R3 0.0 0.0 0.0 0.0 0.0

7-97

Flutter Analysis Sample Problems

Chapter 7

7-98

4

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5

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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

1.252576E-03 -3.500665E-01 1.914446E-03 -3.269584E-01 2.475442E-03 -1.454043E-01 3.376214E-03 1.704016E-01 4.781527E-03 5.180790E-01 6.742940E-03 0.0 0.0 -1.057462E-01 4.717513E-04 -2.762330E-01 1.222850E-03 -3.949783E-01 1.846614E-03 -3.893057E-01 2.539959E-03 -2.115109E-01 3.563229E-03 8.650444E-02 5.139691E-03 4.320273E-01 7.162603E-03 -7.613319E-04 2.032110E-06 -1.725589E-01 4.765802E-04 -3.939156E-01 1.025072E-03 -5.811859E-01 1.746014E-03 -6.031808E-01 2.740918E-03 -4.532699E-01 4.232312E-03 -1.755991E-01 6.219317E-03 1.553009E-01 8.504001E-03 0.0 0.0 -2.165249E-01 2.569513E-04 -5.767798E-01 8.107381E-04 -8.201619E-01 1.663227E-03 -8.754129E-01 3.035255E-03 -7.342857E-01 5.001603E-03 -4.605806E-01 7.384573E-03 -1.330768E-01 9.902165E-03 2.970025E-03 -3.633523E-05 -2.432260E-01 1.613844E-04 -6.529070E-01 7.303757E-04 -9.143163E-01 1.639469E-03 -9.792700E-01 3.152308E-03 -8.333775E-01 5.280297E-03 -5.617344E-01 7.799421E-03 -2.275632E-01 1.037391E-02

1.604266E-03 8.461757E-02 2.333258E-04 6.632644E-02 1.290557E-03 4.847316E-01 8.572310E-04 4.330379E-01 2.182825E-03 6.047416E-01 1.981041E-03 0.0 0.0 -2.063069E-01 1.057373E-03 -1.402375E-01 8.375988E-04 -5.918887E-02 8.736462E-04 1.900489E-01 9.244682E-04 3.823791E-01 1.427989E-03 5.122021E-01 1.970882E-03 5.310512E-01 2.309329E-03 -2.004177E-01 6.397676E-04 -1.862808E-01 5.518922E-04 -2.750368E-01 9.221211E-04 -5.805577E-02 1.027209E-03 1.605820E-01 1.470712E-03 3.807673E-01 1.976250E-03 4.895138E-01 2.386376E-03 5.316419E-01 2.511239E-03 0.0 0.0 -4.755317E-01 6.976659E-04 -3.075665E-01 8.762083E-04 -1.152419E-01 1.407193E-03 1.773268E-01 1.998125E-03 3.760196E-01 2.559543E-03 5.040348E-01 2.710759E-03 5.184990E-01 2.718639E-03 -1.751309E-01 1.330563E-04 -3.895693E-01 5.489420E-04 -4.444935E-01 1.052046E-03 -1.480589E-03 1.386720E-03 8.033113E-02 2.297115E-03 4.936429E-01 2.577077E-03 3.923703E-01 2.929659E-03 6.476611E-01 2.592625E-03

1.460441E-05 1.436280E-01 3.370527E-04 2.455709E-01 -3.039046E-04 2.347115E-01 -6.604888E-04 3.323780E-01 -1.420169E-03 3.207350E-01 -1.562426E-03 0.0 0.0 1.130666E-01 -1.629300E-04 1.088355E-01 2.516959E-04 2.097838E-01 1.386952E-04 2.336975E-01 -1.949048E-04 2.845052E-01 -8.042280E-04 3.192022E-01 -1.336835E-03 3.488120E-01 -1.679773E-03 4.829248E-03 3.248140E-06 5.515521E-02 1.614778E-04 1.920642E-01 2.477137E-04 2.663949E-01 1.078458E-04 3.155970E-01 -3.312527E-04 3.374183E-01 -9.140982E-04 3.558339E-01 -1.427018E-03 3.637772E-01 -1.747523E-03 0.0 0.0 6.698423E-02 3.854881E-04 2.786430E-01 2.881770E-04 3.485368E-01 8.881915E-05 3.843214E-01 -4.315371E-04 3.858066E-01 -1.057288E-03 3.773414E-01 -1.550985E-03 3.780205E-01 -1.820349E-03 -3.174018E-02 1.877310E-04 8.364099E-02 3.924676E-04 3.200474E-01 2.990064E-04 3.592404E-01 1.046184E-04 4.260781E-01 -4.790440E-04 3.754407E-01 -1.068973E-03 4.079929E-01 -1.625045E-03 3.530865E-01 -1.794198E-03

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

Listing 7-28. Output for the 15-Degree Sweptback Wing with Piston Theory Aerodynamics

7.10 Flutter Analysis of a Square Simply Supported Panel (Examples HA145HA and HA145HB) Interest in the flutter of flat and curved panels (plates) began in Germany during World War II as a possible explanation for in-flight failures of V-2 missiles. Current interest is concerned with the design of surface structures of aircraft cruising at high supersonic speeds. The literature on the analysis of fluttering panels and design to prevent their flutter has been surveyed by Rodden (1977). Rectangular flat panels with simple and clamped supports have been studied extensively using various aerodynamic approximations. The assumption of flow on only one side of the panel is realized in NX Nastran by using only half of the air density of the airstream in the flutter analysis. The Piston Theory in NX Nastran is applicable to panels at Mach numbers perhaps as low as m = 1.5, and the ZONA51 supersonic aerodynamics are valid in the low supersonic regime down to a Mach number close to m = 1.0. Two examples are considered here, both of a square panel with simply supported edges at Mach numbers m = 2.0 and 3.0 at sea level and neglecting structural damping. These are Example HA145HA, using Piston Theory, and Example HA145HB, using ZONA51. The configuration is shown in Figure 7-13.

Figure 7-13. Rectangular Panel and Coordinate System The panel is assumed to be square with sides a = b = 10.0 in. and thickness t = 0.041 in. and to be made of aluminum with moduli E = 10.3 × 106 and G = 3.9 × 106 psi and density r = 0.100 lb/in3. The panel is modeled with 100 square CQUAD4 elements as shown in Figure 7-14. The model is contained in input file PANEL_STRUCT.DAT that contains the 121 GRID entries and the 100 CQUAD4 entries, as well as the SPC1 entries required to restrain rigid body rotation, to obtain the simple supports around the edges, and to restrain the drilling degrees of freedom of the QUAD4 elements. PANEL_STRUC.DAT also contains the panel properties on the PSHELL and MAT1 entries; transverse shear effects are neglected. The PARAMs GRDPNT, WTMASS, and COUPMASS are also included. PARAM,WTMASS converts the input weight data to mass data, and PARAM,COUPMASS generates a coupled (not a consistent) mass matrix for the vibration analysis. The structural data are shown in Listing 7-29 and are included in the flutter analyses by the request INCLUDE PANEL_STRUCT.DAT.

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Flutter Analysis Sample Problems

Figure 7-14. Structural Grid Points and QUAD4 Elements Example HA145HA uses Piston Theory and models the panel surface by 10 CAERO5 entries along the chord, each being divided into 10 equal spanwise strips as shown in Figure 7-15. Each CAERO5 entry specifies the one PAERO5, 1001 entry that provides the two Mach numbers, m = 2.0 and 3.0, and zero trim angle of attack data on AEFACT 20. Each CAERO5 entry also specifies the NSPAN = 10 equal strip widths, the Van Dyke correction to Piston Theory by NTHRY = 1, the NTHICK = 10 zero thickness integrals on AEFACT 10, and the spanwise strip geometry with a unit chord. The linear spline SPLINE2 is consistent with the rigid chord assumption of Piston Theory. Ten SPLINE2 entries connect each of the 10 spanwise strips to their respective forward and aft GRIDs via 10 SET1 entries and 10 CORD2R elastic axes through the midchord of each spanwise strip. The flutter analysis entries begin with the AERO entry, which specifies the aerodynamic coordinate system with its positive x-axis streamwise to be the basic coordinate system, the reference chord as 10.0 in., and the sea-level density. The MKAERO1 entry specifies the two Mach numbers m = 2.0 and 3.0 and the four reduced frequencies, k = 0.001, 0.1, 0.2, and 0.4, for the generation of the Piston Theory aerodynamic influence coefficients. The FLUTTER entry specifies the PK-method and requests four modes output from the flutter analysis. FLFACT 1 introduces the factor of 0.5 on the density to account for the air only flowing on one side of the panel. FLFACT 2 specifies the two Mach numbers. FLFACT 3 specifies the velocities for the PK-flutter analysis in in/s and PARAM,VREF,12.0 provides the

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factor to output the flutter speeds in the Flutter Summary table in units of ft/s. The vibration analysis method is specified by the EIGR entry as the Givens method and requests 15 modes to be normalized on their largest components. All 15 modes are used in the flutter analysis. Finally, PARAM,OPPHIPA, along with a DISP request in Case Control, provides the deflections of all grid points including the aerodynamic points which, for Piston Theory, are located at the quarter-chord of the centerline of each aerodynamic box. It should be mentioned that the flutter mode of the square panel is symmetric about its streamwise centerline and that the 15 modes selected include both symmetric and antisymmetric modes. It is not likely that the 10 × 10 CQUAD4 model of the panel structure will have an accurate 15th mode, but the correlation shown below indicates that it is a reasonable model.

Case Control Commands The Case Control Section includes the title, subtitle, label, ECHO = BOTH, METHOD = 20 for the vibration analysis, and FMETHOD = 30 for the PK flutter analysis. SPC = 1 includes all of the Bulk Data constraints. DISP = 10 and SET 10 = 1 thru 999,999 requests the vibration modes at all grid points including the aerodynamic grid points. BEGIN BULK concludes the Case Control Section. The Executive Control Section includes ID NXN, HA145HA, TIME 25 minutes, and SOL 145, and concludes with CEND.

Output The input data are shown in Listing 7-30 and Listing 7-31. Selected output data are shown in Listing 7-32 and are discussed below. The output from the GRID POINT WEIGHT GENERATOR checks some of the structural data input with the weight of 0.410 lbs and centroid at x = y = 5.0 in. The first vibration frequency is f1 = 78.57 Hz. This agrees with the theoretical value, with transverse shear effects neglected, of where the plate stiffness D = Et3/12(1 − v2) = 65.93 in-lb based on Poisson’s ratio v = E/2G − 1 = 0.3205. The second theoretical frequency is and corresponds to a calculated value of 200.43 Hz, and the limitations of the 10 × 10 element structural model are apparent in the loss in accuracy of the calculated second frequency. Although 15 modes are used in the flutter analysis, the flutter is caused primarily by the coupling between the first two symmetric modes and great accuracy is not required in the higher modes. The first three vibration modes follow the frequency output. Since the second and third modes have identical frequencies, any linear combination of the second and third modes shown are also modes. Appropriate combinations of these modes would correspond to the second symmetric and second antisymmetric modes. The flutter data summaries for the first four modes are shown next, and the dampings and frequencies are plotted in Figure 7-16 for the critical second mode at the two Mach numbers. At a Mach number m = 2.0 the flutter speed is V = 2436 ft/s, and at m = 3.0 the speed is V = 2436 ft/s; the flutter frequencies are 171 and 173 Hz, respectively. The nondimensional stability where β2 = M2 − 1. The parameter customarily used in panel flutter studies is values of λ for the two Mach numbers are found to be λcr = 532 at m = 2.0 and λcr = 526 at m = 3.0. These compare reasonably well with the theoretical values given by Hedgepeth (1957, Table 2) of λcr = 492 at m = 2.0 and λcr = 499 at m = 3.0.

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Figure 7-15. CAERO5 Entries for 10 Chordwise PanelsEach Divided into 10 Spanwise Strips

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Figure 7-16. Critical Flutter V-g and V-f Curves for Piston Theory

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Example HA145HB uses ZONA51 supersonic aerodynamics and models the panel surface by a single CAERO1 entry divided into 100 equal boxes. It also models the regions to either side of the panel by square CAERO1 panels that have no motion so that the center panel behaves as a panel within a larger surface rather than as a wing. The lateral CAERO1 panels prevent the edges of the central CAERO1 panel from acting like tips on an isolated wing. The aspect ratio of the lateral panels must be such that the Mach lines from the forward outboard corners do not intersect the side edges of the central panel as shown in Figure 7-17. The square lateral . The Bulk panels chosen here would permit an analysis Mach number as low as Data Section is similar to that described above for Piston Theory. The three CAERO1 entries specify the panel and its lateral counterparts, each divided into 10 equal spanwise and 10 equal chordwise boxes. The PAERO1 entry is required. The 10 spanwise linear splines are the same as those required to obtain accurate interpolation at the aerodynamic grid points for Piston Theory; note that Remark 1 on the SPLINE1 entry also applies to the SPLINE2 entry and the box numbering for each SPLINE2 is taken from Figure 7-18. The SET1 and CORD2R entries are those used in the Piston Theory analysis. The remainder of the Bulk Data entries are the same as in the Piston Theory analysis.

Figure 7-17. Mach Wave Interference from Lateral Panels

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Figure 7-18. CAERO1 Box Numbering for ZONA51 Supersonic Aerodynamics The input data are shown in Listing 7-33 and Listing 7-34 and the Flutter Summary tables are shown in Listing 7-35. The dampings and frequencies are plotted in Figure 7-19, again for the

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critical second mode at the two Mach numbers. At m = 2.0 the flutter speed is V = 1873 ft/s, and at m = 3.0 the speed is V = 2410 ft/s; the flutter frequencies are 169 and 173 Hz, respectively. The values of the stability parameter are λcr = 507 at m = 2.0 and λcr = 515 at m = 3.0. These compare reasonably well with the Piston Theory results above and more closely with the values of Hedgepeth (1957, Table 2). ZONA51 appears to be more conservative than Piston Theory. The panel stability parameters are compared in Table 7-2 for the various methods at the two Mach numbers. One would expect better agreement between Piston Theory and ZONA51 at the higher Mach number, and this is the case.

Table 7-2. The Panel Stability Parameter λ Piston Theory Mach Number 2.0 532

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cr

ZONA51 507

Hedgepeth (1957) 492

Flutter Analysis Sample Problems

cr

Table 7-2. The Panel Stability Parameter λ Piston Theory Mach Number 3.0 526

ZONA51 515

Hedgepeth (1957) 499

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$ IT CAN BE USED HERE FOR GRID POINT GENERATION AND QUAD4 $ GEOMETRY GENERATION BECAUSE OF THE UNIFORM DIVISIONS OF $ MODEL IN THIS EXAMPLE. $ GRID 1 0.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 12 1.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 23 2.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 34 3.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 45 4.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 56 5.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 67 6.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 78 7.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 89 8.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 100 9.0 0.0 0.0 = *1 = = *1.0 = =9 GRID 111 10.0 0.0 0.0 = *1 = = *1.0 = =9 $ CQUAD4 1 1 1 2 13 12 = *1 = *1 *1 *1 *1 =8 CQUAD4 11 1 12 13 24 23 = *1 = *1 *1 *1 *1 =8 CQUAD4 21 1 23 24 35 34 = *1 = *1 *1 *1 *1 =8 CQUAD4 31 1 34 35 46 45 = *1 = *1 *1 *1 *1 =8 CQUAD4 41 1 45 46 57 56 = *1 = *1 *1 *1 *1 =8 CQUAD4 51 1 56 57 68 67 = *1 = *1 *1 *1 *1 =8 CQUAD4 61 1 67 68 79 78 = *1 = *1 *1 *1 *1 =8 CQUAD4 71 1 78 79 90 89 = *1 = *1 *1 *1 *1 =8 CQUAD4 81 1 89 90 101 100 = *1 = *1 *1 *1 *1 =8 CQUAD4 91 1 100 101 112 111 = *1 = *1 *1 *1 *1 =8 $ SPC1 1 1 1 SPC1 1 2 1 110 SPC1 1 3 1 THRU 11 SPC1 1 3 111 THRU 121 SPC1 1 3 12 22 23 33 34 +SP1 45 55 56 66 67 77 78 +SP2 89 99 100 110 SPC1 1 6 1 THRU 121 $ $ PROPERTY AND MATERIAL BULK DATA ENTRIES $ PSHELL 1 10 .041 10 $ MAT1 10 1.03+7 3.9+6 .1 $ PARAM GRDPNT 1 $ PARAM WTMASS .0025907 $ PARAM COUPMASS1 $

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ELEMENT THE STRUCTURAL

44 88

+SP1 +SP2

Flutter Analysis Sample Problems

Listing 7-29. Bulk Data Entries for PANEL_STRUCT.DAT Input File N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA145HA TIME 25 SOL 145 CEND 3D PANEL FLUTTER DECEMBER 17, 2004 NX NASTRAN 12/15/04 PAGE PINNED EDGES PISTON THEORY AERO WITH VAN DYKE CORRECTION, EXAMPLE HA145HA C A S E C O N T R O L D E C K E C H O CARD COUNT 1 $ 2 TITLE = 3D PANEL FLUTTER 3 SUBTI = PINNED EDGES 4 LABEL = PISTON THEORY AERO WITH VAN DYKE CORRECTION, EXAMPLE HA145HA 5 ECHO = BOTH 6 METHOD = 20 7 FMETHOD = 30 8 SPC = 1 9 SET 10 = 1 THRU 999999 10 DISP = 10 11 BEGIN BULK 3D PANEL FLUTTER PAGE 3 PINNED EDGES PISTON THEORY AERO WITH VAN DYKE CORRECTION, EXAMPLE HA145HA I N P U T B U L K D A T A D E C K E C H O .

1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 $ INCLUDE PANEL_STRUCT.DAT $ $ TEN CAERO5 ENTRIES FOR PANEL SURFACE $ $ EID PID CP NSPAN LSPAN NTHRY NTHICK CAERO5 1001 1001 10 1 10 $ X1 Y1 Z1 X12 X4 Y4 Z4 X43 +CA501 0.0 0.0 0.0 1.0 0.0 10.0 0.0 1.0 $ CAERO5 2001 1001 10 1 10 +CA502 1.0 0.0 0.0 1.0 1.0 10.0 0.0 1.0 CAERO5 3001 1001 10 1 10 +CA503 2.0 0.0 0.0 1.0 2.0 10.0 0.0 1.0 CAERO5 4001 1001 10 1 10 +CA504 3.0 0.0 0.0 1.0 3.0 10.0 0.0 1.0 CAERO5 5001 1001 10 1 10 +CA505 4.0 0.0 0.0 1.0 4.0 10.0 0.0 1.0 CAERO5 6001 1001 10 1 10 +CA506 5.0 0.0 0.0 1.0 5.0 10.0 0.0 1.0 CAERO5 7001 1001 10 1 10 +CA507 6.0 0.0 0.0 1.0 6.0 10.0 0.0 1.0 CAERO5 8001 1001 10 1 10 +CA508 7.0 0.0 0.0 1.0 7.0 10.0 0.0 1.0 CAERO5 9001 1001 10 1 10 +CA509 8.0 0.0 0.0 1.0 8.0 10.0 0.0 1.0 CAERO5 10001 1001 10 1 10 +CA510 9.0 0.0 0.0 1.0 9.0 10.0 0.0 1.0 $ $ THICKNESS INTEGRALS AEFACT 10 0.0 0.0 0.0 0.0 0.0 0.0 $ $ PID NALPHA LALPHA NXIS LXIS NTAUS LTAUS PAERO5 1001 1 20 $ CAOC1 CAOC2 ETC +PA5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 +PA51 0.0 0.0 $ $ MACH, ALPHA COMBOS ALPHA21 ETC $ SID MACH1 ALPHA11 MACH2 AEFACT 20 2.0 0.0 3.0 0.0 $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID 1001 0.0 1.0 1 SPLINE2 1001 1001 1001 1010 $+SPL DTHX DTHY +SPL1 -1.0 -1.0 $ 2001 0.0 1.0 2 SPLINE2 2001 2001 2001 2010 +SPL2 -1.0 -1.0 3001 0.0 1.0 3 SPLINE2 3001 3001 3001 3010 +SPL3 -1.0 -1.0 4001 0.0 1.0 4 SPLINE2 4001 4001 4001 4010 +SPL4 -1.0 -1.0 5001 0.0 1.0 5 SPLINE2 5001 5001 5001 5010 +SPL5 -1.0 -1.0

..

10

2

.

+CA501

+CA502 +CA503 +CA504 +CA505 +CA506 +CA507 +CA508 +CA509 +CA510

+PA5 +PA51

+SPL +SPL1

+SPL2 +SPL3 +SPL4 +SPL5

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Chapter 7

SPLINE2 6001 +SPL6 -1.0 SPLINE2 7001 +SPL7 -1.0 SPLINE2 8001 +SPL8 -1.0 SPLINE2 9001 +SPL9 -1.0 SPLINE2 10001 +SPL10 -1.0 $ $ SID SET1 1001 SET1 2001 SET1 3001 SET1 4001 SET1 5001 SET1 6001 SET1 7001 SET1 8001 SET1 9001 SET1 10001 $ $ CID CORD2R 1 $ C1 +CRD1 20.0 CORD2R 2 +CRD2 20.0 CORD2R 3 +CRD3 20.0 CORD2R 4 +CRD4 20.0 CORD2R 5 +CRD5 20.0 CORD2R 6 +CRD6 20.0 CORD2R 7 +CRD7 20.0 CORD2R 8 +CRD8 20.0 CORD2R 9 +CRD9 20.0 CORD2R 10 +CRD10 20.0 $ AERO 0 $ $ M1 MKAERO1 2.0 $ K1 +MK 0.001 $ FLUTTER 30 FLFACT 1 FLFACT 2 FLFACT 3 +FL3 30000. $ PARAM VREF $ EIGR 20 +ER MAX $ $PARAM LMODES $ PARAM OPPHIPA $ ENDDATA

6001 -1.0 7001 -1.0 8001 -1.0 9001 -1.0 10001 -1.0

6001

6010

6001

0.0

1.0

6

+SPL6

7001

7010

7001

0.0

1.0

7

+SPL7

8001

8010

8001

0.0

1.0

8

+SPL8

9001

9010

9001

0.0

1.0

9

+SPL9

10001

10010

10001

0.0

1.0

10

+SPL10

G1 1 12 23 34 45 56 67 78 89 100

G2 THRU THRU THRU THRU THRU THRU THRU THRU THRU THRU

G3 22 33 44 55 66 77 88 99 110 121

ETC

RID

A2 0.0

A3 0.0

B1 0.5

B2 0.0

B3 1.0

+CRD +CRD1

0.0

0.0

1.5

0.0

1.0

+CRD2

0.0

0.0

2.5

0.0

1.0

+CRD3

0.0

0.0

3.5

0.0

1.0

+CRD4

0.0

0.0

4.5

0.0

1.0

+CRD5

0.0

0.0

5.5

0.0

1.0

+CRD6

0.0

0.0

6.5

0.0

1.0

+CRD7

0.0

0.0

7.5

0.0

1.0

+CRD8

0.0

0.0

8.5

0.0

1.0

+CRD9

0.0

0.0

9.5

0.0

1.0

+CRD10

0.0

A1 0.5 C3 0.0 1.5 0.0 2.5 0.0 3.5 0.0 4.5 0.0 5.5 0.0 6.5 0.0 7.5 0.0 8.5 0.0 9.5 0.0

0

10.0

1.1468-7

M2 3.0 K2 0.1

K3 0.2

ETC 0.4

1

2

3

3.0 22800.

23400.

24000.

0.

1000.

C2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

PK 0.5 2.0 21600. 31200.

ETC

+MK +MK

4 26400.

28800.

29400.

+FL3

12.0 AGIV

ND 15

+ER

9 1 INPUT BULK DATA CARD COUNT =

216

Listing 7-30. Input Data for Flutter Analysis of Square Panel Using Piston Theory

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3D PANEL FLUTTER PAGE 9 PINNED EDGES PISTON THEORY AERO WITH VAN DYKE CORRECTION, EXAMPLE HA145HA S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 1AEFACT 10 0.0 0.0 0.0 0.0 0.0 0.0 2AEFACT 20 2.0 0.0 3.0 0.0 3AERO 0 0 10.0 1.1468-7 4CAERO5 1001 1001 10 1 10 5+CA501 0.0 0.0 0.0 1.0 0.0 10.0 0.0 1.0 6CAERO5 2001 1001 10 1 10 7+CA502 1.0 0.0 0.0 1.0 1.0 10.0 0.0 1.0 8CAERO5 3001 1001 10 1 10 9+CA503 2.0 0.0 0.0 1.0 2.0 10.0 0.0 1.0 10CAERO5 4001 1001 10 1 10 11+CA504 3.0 0.0 0.0 1.0 3.0 10.0 0.0 1.0 12CAERO5 5001 1001 10 1 10 13+CA505 4.0 0.0 0.0 1.0 4.0 10.0 0.0 1.0 14CAERO5 6001 1001 10 1 10 15+CA506 5.0 0.0 0.0 1.0 5.0 10.0 0.0 1.0 16CAERO5 7001 1001 10 1 10 17+CA507 6.0 0.0 0.0 1.0 6.0 10.0 0.0 1.0 18CAERO5 8001 1001 10 1 10 19+CA508 7.0 0.0 0.0 1.0 7.0 10.0 0.0 1.0 20CAERO5 9001 1001 10 1 10 21+CA509 8.0 0.0 0.0 1.0 8.0 10.0 0.0 1.0 22CAERO5 10001 1001 10 1 10 23+CA510 9.0 0.0 0.0 1.0 9.0 10.0 0.0 1.0 24CORD2R 1 0.5 0.0 0.0 0.5 0.0 1.0 25+CRD1 20.0 0.0 0.0 26CORD2R 2 1.5 0.0 0.0 1.5 0.0 1.0 27+CRD2 20.0 0.0 0.0 28CORD2R 3 2.5 0.0 0.0 2.5 0.0 1.0 29+CRD3 20.0 0.0 0.0 30CORD2R 4 3.5 0.0 0.0 3.5 0.0 1.0 31+CRD4 20.0 0.0 0.0 32CORD2R 5 4.5 0.0 0.0 4.5 0.0 1.0 33+CRD5 20.0 0.0 0.0 34CORD2R 6 5.5 0.0 0.0 5.5 0.0 1.0 35+CRD6 20.0 0.0 0.0 36CORD2R 7 6.5 0.0 0.0 6.5 0.0 1.0 37+CRD7 20.0 0.0 0.0 38CORD2R 8 7.5 0.0 0.0 7.5 0.0 1.0 39+CRD8 20.0 0.0 0.0 40CORD2R 9 8.5 0.0 0.0 8.5 0.0 1.0 41+CRD9 20.0 0.0 0.0 42CORD2R 10 9.5 0.0 0.0 9.5 0.0 1.0 43+CRD10 20.0 0.0 0.0 44CQUAD4 1 1 1 2 13 12 45CQUAD4 2 1 2 3 14 13 46CQUAD4 3 1 3 4 15 14 47CQUAD4 4 1 4 5 16 15 48CQUAD4 5 1 5 6 17 16 49CQUAD4 6 1 6 7 18 17 50CQUAD4 7 1 7 8 19 18 51CQUAD4 8 1 8 9 20 19 52CQUAD4 9 1 9 10 21 20 53CQUAD4 10 1 10 11 22 21 54CQUAD4 11 1 12 13 24 23 55CQUAD4 12 1 13 14 25 24 56CQUAD4 13 1 14 15 26 25 57CQUAD4 14 1 15 16 27 26 58CQUAD4 15 1 16 17 28 27 59CQUAD4 16 1 17 18 29 28 60CQUAD4 17 1 18 19 30 29 61CQUAD4 18 1 19 20 31 30 62CQUAD4 19 1 20 21 32 31 63CQUAD4 20 1 21 22 33 32 64CQUAD4 21 1 23 24 35 34 65CQUAD4 22 1 24 25 36 35 66CQUAD4 23 1 25 26 37 36 67CQUAD4 24 1 26 27 38 37 68CQUAD4 25 1 27 28 39 38 69CQUAD4 26 1 28 29 40 39 70CQUAD4 27 1 29 30 41 40 71CQUAD4 28 1 30 31 42 41 72CQUAD4 29 1 31 32 43 42 73CQUAD4 30 1 32 33 44 43 74CQUAD4 31 1 34 35 46 45 75CQUAD4 32 1 35 36 47 46 76CQUAD4 33 1 36 37 48 47 77CQUAD4 34 1 37 38 49 48 78CQUAD4 35 1 38 39 50 49 79CQUAD4 36 1 39 40 51 50 80CQUAD4 37 1 40 41 52 51 81CQUAD4 38 1 41 42 53 52 82CQUAD4 39 1 42 43 54 53 83CQUAD4 40 1 43 44 55 54 84CQUAD4 41 1 45 46 57 56 85CQUAD4 42 1 46 47 58 57

..

10

.

+CA501 +CA502 +CA503 +CA504 +CA505 +CA506 +CA507 +CA508 +CA509 +CA510 +CRD1 +CRD2 +CRD3 +CRD4 +CRD5 +CRD6 +CRD7 +CRD8 +CRD9 +CRD10

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Flutter Analysis Sample Problems

8687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158GRID 160161162163164165166167168169170171172173174175176-

CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 EIGR +ER FLFACT FLFACT FLFACT +FL3 FLUTTER GRID GRID GRID GRID GRID GRID GRID GRID 9 GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 20 MAX 1 2 3 30000. 30 1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 AGIV

47 48 49 50 51 52 53 54 56 57 58 59 60 61 62 63 64 65 67 68 69 70 71 72 73 74 75 76 78 79 80 81 82 83 84 85 86 87 89 90 91 92 93 94 95 96 97 98 100 101 102 103 104 105 106 107 108 109 0.

0.5 2.0 21600. 31200. PK

0.0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

NX Nastran Aeroelastic Analysis User’s Guide

48 49 50 51 52 53 54 55 57 58 59 60 61 62 63 64 65 66 68 69 70 71 72 73 74 75 76 77 79 80 81 82 83 84 85 86 87 88 90 91 92 93 94 95 96 97 98 99 101 102 103 104 105 106 107 108 109 110 1000.

59 60 61 62 63 64 65 66 68 69 70 71 72 73 74 75 76 77 79 80 81 82 83 84 85 86 87 88 90 91 92 93 94 95 96 97 98 99 101 102 103 104 105 106 107 108 109 110 112 113 114 115 116 117 118 119 120 121

3.0 22800.

23400.

24000.

1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8. 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0

2 0.0 1.0 2. 3. 4. 5. 6. 7. 0.0 9. 10. 0.0 1.0 2. 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3.

3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

58 59 60 61 62 63 64 65 67 68 69 70 71 72 73 74 75 76 78 79 80 81 82 83 84 85 86 87 89 90 91 92 93 94 95 96 97 98 100 101 102 103 104 105 106 107 108 109 111 112 113 114 115 116 117 118 119 120 15

26400.

+ER

28800. 4

29400.

+FL3

Flutter Analysis Sample Problems

177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241-

GRID 242243244245246247248249250251252253254255256257258259260261262263264265266267-

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID 91 GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 8.0 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117

2.0 2.0 2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 8.0 8.0 2. 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 0.0 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3. 4. 5. 6. 7. 8. 9. 10. 0.0 1.0 2. 3. 4. 5. 6.

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321-

7-114

GRID GRID GRID GRID MAT1 MKAERO1 +MK PAERO5 +PA5 +PA51 PARAM PARAM PARAM PARAM PARAM PSHELL SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SPC1 SPC1 SPC1 SPC1 +SP1 +SP2 SPC1 SPC1 SPLINE2 +SPL1 SPLINE2 +SPL2 SPLINE2 +SPL3 SPLINE2 +SPL4 SPLINE2 +SPL5 SPLINE2 +SPL6 SPLINE2 +SPL7 SPLINE2 +SPL8 SPLINE2 +SPL9 SPLINE2 +SPL10 ENDDATA TOTAL COUNT=

118 10.0 119 10.0 120 10.0 121 10.0 10 1.03+7 3.9+6 2.0 3.0 0.001 0.1 0.2 1001 1 20 0.0 0.0 0.0 0.0 0.0 COUPMASS1 GRDPNT 1 OPPHIPA 1 VREF 12.0 WTMASS .0025907 1 10 .041 1001 1 THRU 2001 12 THRU 3001 23 THRU 4001 34 THRU 5001 45 THRU 6001 56 THRU 7001 67 THRU 8001 78 THRU 9001 89 THRU 10001 100 THRU 1 1 1 1 2 1 1 3 1 1 3 12 45 55 56 89 99 100 1 3 111 1 6 1 1001 1001 1001 -1.0 -1.0 2001 2001 2001 -1.0 -1.0 3001 3001 3001 -1.0 -1.0 4001 4001 4001 -1.0 -1.0 5001 5001 5001 -1.0 -1.0 6001 6001 6001 -1.0 -1.0 7001 7001 7001 -1.0 -1.0 8001 8001 8001 -1.0 -1.0 9001 9001 9001 -1.0 -1.0 10001 10001 10001 -1.0 -1.0 322

NX Nastran Aeroelastic Analysis User’s Guide

7. 8. 9. 10.

0.0 0.0 0.0 0.0 .1 +MK

0.4 0.0

0.0

0.0

0.0

0.0

+PA5 +PA51

10 22 33 44 55 66 77 88 99 110 121 110 THRU 22 66 110 THRU THRU 1010

11 23 67

33 77

34 78

44 88

+SP1 +SP2

121 121 1001

0.0

1.0

1

+SPL1

2010

2001

0.0

1.0

2

+SPL2

3010

3001

0.0

1.0

3

+SPL3

4010

4001

0.0

1.0

4

+SPL4

5010

5001

0.0

1.0

5

+SPL5

6010

6001

0.0

1.0

6

+SPL6

7010

7001

0.0

1.0

7

+SPL7

8010

8001

0.0

1.0

8

+SPL8

9010

9001

0.0

1.0

9

+SPL9

10010

10001

0.0

1.0

10

+SPL10

Flutter Analysis Sample Problems

Listing 7-31. Sorted Bulk Data for Flutter Analysis of Square Panel Using Piston Theory 3D PANEL FLUTTER PAGE 19 PINNED EDGES PISTON THEORY AERO WITH VAN DYKE CORRECTION, EXAMPLE HA145HA O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 1 M O * 4.100000E-01 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -2.050000E+00 * * 0.000000E+00 4.100000E-01 0.000000E+00 0.000000E+00 0.000000E+00 2.050000E+00 * * 0.000000E+00 0.000000E+00 4.100000E-01 2.050000E+00 -2.050000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 2.050000E+00 1.366667E+01 -1.025000E+01 0.000000E+00 * * 0.000000E+00 0.000000E+00 -2.050000E+00 -1.025000E+01 1.366667E+01 0.000000E+00 * * -2.050000E+00 2.050000E+00 0.000000E+00 0.000000E+00 0.000000E+00 2.733333E+01 * S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 4.100000E-01 0.000000E+00 5.000000E+00 0.000000E+00 Y 4.100000E-01 5.000000E+00 0.000000E+00 0.000000E+00 Z 4.100000E-01 5.000000E+00 5.000000E+00 0.000000E+00 I(S) * 3.416665E+00 -1.907349E-06 0.000000E+00 * * -1.907349E-06 3.416665E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 6.833330E+00 * I(Q) * 3.416665E+00 * * 3.416665E+00 * * 6.833330E+00 * Q * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 303 2.437391E+05 4.936994E+02 7.857470E+01 2.569396E-04 6.262624E+01 2 318 1.585938E+06 1.259340E+03 2.004302E+02 2.385697E-04 3.783567E+02 3 317 1.585938E+06 1.259340E+03 2.004302E+02 2.385697E-04 3.783567E+02 4 304 4.074306E+06 2.018491E+03 3.212528E+02 2.844406E-04 1.158898E+03 5 309 6.896249E+06 2.626071E+03 4.179522E+02 1.134159E-04 7.821444E+02 6 310 6.896309E+06 2.626083E+03 4.179540E+02 1.902268E-04 1.311863E+03 7 319 1.147131E+07 3.386933E+03 5.390471E+02 1.608912E-04 1.845633E+03 8 308 1.147131E+07 3.386933E+03 5.390471E+02 1.608912E-04 1.845633E+03 9 320 2.243252E+07 4.736298E+03 7.538052E+02 2.058151E-04 4.616952E+03 10 313 2.243252E+07 4.736298E+03 7.538052E+02 2.058151E-04 4.616952E+03 11 307 2.257317E+07 4.751123E+03 7.561647E+02 1.957289E-04 4.418221E+03 12 305 3.065322E+07 5.536535E+03 8.811669E+02 2.404108E-04 7.369367E+03 13 306 3.065704E+07 5.536880E+03 8.812218E+02 2.445093E-04 7.495932E+03 14 302 4.779749E+07 6.913573E+03 1.100329E+03 1.659348E-04 7.931270E+03 15 316 4.779749E+07 6.913573E+03 1.100329E+03 1.659348E-04 7.931270E+03 EIGENVALUE = 2.437391E+05 CYCLES = 7.857470E+01 POINT ID.

TYPE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R E A L T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

E I G E N V E C T O R T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 9.547547E-02 1.816133E-01 2.499747E-01 2.938657E-01 3.089895E-01 2.938657E-01 2.499747E-01 1.816133E-01 9.547547E-02 0.0 0.0 1.816133E-01 3.454662E-01 4.755045E-01 5.589946E-01 5.877633E-01 5.589946E-01 4.755045E-01 3.454662E-01

N O .

R1 8.264631E-04 7.895400E-04 6.727834E-04 4.893358E-04 2.573338E-04 2.115821E-15 -2.573338E-04 -4.893358E-04 -6.727834E-04 -7.895400E-04 -8.264631E-04 9.712248E-02 9.238406E-02 7.859124E-02 5.710068E-02 3.001965E-02 -4.522600E-14 -3.001965E-02 -5.710068E-02 -7.859124E-02 -9.238406E-02 -9.712248E-02 1.841179E-01 1.751367E-01 1.489877E-01 1.082471E-01 5.690884E-02 -8.636837E-14 -5.690884E-02 -1.082471E-01 -1.489877E-01

1 R2 -8.264631E-04 -9.712248E-02 -1.841179E-01 -2.532092E-01 -2.975848E-01 -3.128780E-01 -2.975848E-01 -2.532092E-01 -1.841179E-01 -9.712248E-02 -8.264631E-04 -7.895400E-04 -9.238406E-02 -1.751367E-01 -2.408578E-01 -2.830689E-01 -2.976160E-01 -2.830689E-01 -2.408578E-01 -1.751367E-01 -9.238406E-02 -7.895400E-04 -6.727834E-04 -7.859124E-02 -1.489877E-01 -2.048963E-01 -2.408050E-01 -2.531802E-01 -2.408050E-01 -2.048963E-01 -1.489877E-01

NX Nastran Aeroelastic Analysis User’s Guide

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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Flutter Analysis Sample Problems

Chapter 7

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85

7-116

G 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

1.816133E-01 0.0 0.0 2.499747E-01 4.755045E-01 6.544914E-01 7.694086E-01 8.090063E-01 7.694086E-01 6.544914E-01 4.755045E-01 2.499747E-01 0.0 0.0 2.938657E-01 5.589946E-01 7.694086E-01 9.045033E-01 9.510538E-01 9.045033E-01 7.694086E-01 5.589946E-01 2.938657E-01 0.0 0.0 3.089895E-01 5.877633E-01 8.090063E-01 9.510538E-01 1.000000E+00 9.510538E-01 8.090063E-01 5.877633E-01 3.089895E-01 0.0 0.0 2.938657E-01 5.589946E-01 7.694086E-01 9.045033E-01 9.510538E-01 9.045033E-01 7.694086E-01 5.589946E-01 2.938657E-01 0.0 0.0 2.499747E-01 4.755045E-01 6.544914E-01 7.694086E-01 8.090063E-01 7.694086E-01

-1.751367E-01 -1.841179E-01 2.532092E-01 2.408578E-01 2.048963E-01 1.488674E-01 7.826415E-02 -1.219637E-13 -7.826415E-02 -1.488674E-01 -2.048963E-01 -2.408578E-01 -2.532092E-01 2.975848E-01 2.830689E-01 2.408050E-01 1.749569E-01 9.198016E-02 -1.531339E-13 -9.198016E-02 -1.749569E-01 -2.408050E-01 -2.830689E-01 -2.975848E-01 3.128780E-01 2.976160E-01 2.531802E-01 1.839480E-01 9.670710E-02 -1.692418E-13 -9.670710E-02 -1.839480E-01 -2.531802E-01 -2.976160E-01 -3.128780E-01 2.975848E-01 2.830689E-01 2.408050E-01 1.749569E-01 9.198016E-02 -1.703268E-13 -9.198016E-02 -1.749569E-01 -2.408050E-01 -2.830689E-01 -2.975848E-01 2.532092E-01 2.408578E-01 2.048963E-01 1.488674E-01 7.826415E-02 -1.458745E-13 -7.826415E-02

-7.859124E-02 -6.727834E-04 -4.893358E-04 -5.710068E-02 -1.082471E-01 -1.488674E-01 -1.749569E-01 -1.839480E-01 -1.749569E-01 -1.488674E-01 -1.082471E-01 -5.710068E-02 -4.893358E-04 -2.573338E-04 -3.001965E-02 -5.690884E-02 -7.826415E-02 -9.198016E-02 -9.670710E-02 -9.198016E-02 -7.826415E-02 -5.690884E-02 -3.001965E-02 -2.573338E-04 4.657190E-16 -2.972403E-14 -5.555368E-14 -6.635859E-14 -6.528196E-14 -5.969057E-14 -4.617972E-14 -3.487857E-14 -3.507687E-14 -2.423626E-14 -7.589415E-17 2.573338E-04 3.001965E-02 5.690884E-02 7.826415E-02 9.198016E-02 9.670710E-02 9.198016E-02 7.826415E-02 5.690884E-02 3.001965E-02 2.573338E-04 4.893358E-04 5.710068E-02 1.082471E-01 1.488674E-01 1.749569E-01 1.839480E-01 1.749569E-01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

6.544914E-01 -1.488674E-01 1.488674E-01 0.0 4.755045E-01 -2.048963E-01 1.082471E-01 2.499747E-01 -2.408578E-01 5.710068E-02 0.0 -2.532092E-01 4.893358E-04 0.0 1.841179E-01 6.727834E-04 1.816133E-01 1.751367E-01 7.859124E-02 3.454662E-01 1.489877E-01 1.489877E-01 4.755045E-01 1.082471E-01 2.048963E-01 5.589946E-01 5.690884E-02 2.408050E-01 5.877633E-01 -9.868105E-14 2.531802E-01 5.589946E-01 -5.690884E-02 2.408050E-01 4.755045E-01 -1.082471E-01 2.048963E-01 3.454662E-01 -1.489877E-01 1.489877E-01 1.816133E-01 -1.751367E-01 7.859124E-02 0.0 -1.841179E-01 6.727834E-04 0.0 9.712248E-02 7.895400E-04 9.547547E-02 9.238406E-02 9.238406E-02 1.816133E-01 7.859124E-02 1.751367E-01 2.499747E-01 5.710068E-02 2.408578E-01 2.938657E-01 3.001965E-02 2.830689E-01 3.089895E-01 -4.968942E-14 2.976160E-01 2.938657E-01 -3.001965E-02 2.830689E-01 2.499747E-01 -5.710068E-02 2.408578E-01 1.816133E-01 -7.859124E-02 1.751367E-01 9.547547E-02 -9.238406E-02 9.238406E-02 0.0 -9.712248E-02 7.895400E-04 0.0 8.264631E-04 8.264631E-04 0.0 7.895400E-04 9.712248E-02 0.0 6.727834E-04 1.841179E-01 0.0 4.893358E-04 2.532092E-01 0.0 2.573338E-04 2.975848E-01 0.0 3.291204E-15 3.128780E-01 0.0 -2.573338E-04 2.975848E-01 0.0 -4.893358E-04 2.532092E-01 0.0 -6.727834E-04 1.841179E-01 0.0 -7.895400E-04 9.712248E-02 0.0 -8.264631E-04 8.264631E-04

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 3001 3002 3003 3004 3005 3006 3007 3008 3009 3010 4001 4002 4003 4004 4005 4006 4007 4008 4009 4010 5001

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

5002 5003 5004 5005 5006 5007 5008 5009 5010 6001 6002 6003 6004 6005 6006 6007 6008 6009 6010 7001 7002 7003 7004 7005 7006 7007 7008 7009 7010 8001 8002 8003 8004 8005 8006 8007 8008 8009 8010 9001 9002 9003 9004 9005 9006 9007 9008 9009 9010 10001

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

1.223094E-02 3.549739E-02 5.529068E-02 6.967133E-02 7.723170E-02 7.723170E-02 6.967132E-02 5.529068E-02 3.549739E-02 1.223094E-02 5.936538E-02 1.722928E-01 2.683622E-01 3.381610E-01 3.748565E-01 3.748565E-01 3.381609E-01 2.683622E-01 1.722928E-01 5.936538E-02 1.006919E-01 2.922330E-01 4.551809E-01 5.735700E-01 6.358110E-01 6.358110E-01 5.735699E-01 4.551808E-01 2.922330E-01 1.006919E-01 1.321623E-01 3.835681E-01 5.974443E-01 7.528353E-01 8.345295E-01 8.345295E-01 7.528352E-01 5.974442E-01 3.835681E-01 1.321623E-01 1.506951E-01 0.0 4.373549E-01 6.812225E-01 8.584037E-01 9.515538E-01 9.515538E-01 8.584037E-01 6.812224E-01 4.373549E-01 1.506951E-01 1.544761E-01 4.483280E-01 6.983141E-01 8.799408E-01 9.754279E-01 9.754279E-01 8.799407E-01 6.983141E-01 4.483280E-01 1.544761E-01 1.431350E-01 4.154134E-01 6.470461E-01 8.153383E-01 9.038150E-01 9.038150E-01 8.153383E-01 6.470461E-01 4.154134E-01 1.431350E-01 1.177823E-01 3.418329E-01 5.324371E-01 6.709202E-01 7.437253E-01 7.437253E-01 6.709201E-01 5.324371E-01 3.418329E-01 1.177823E-01 8.089983E-02 2.347905E-01 3.657078E-01 4.608257E-01 5.108322E-01 5.108322E-01 4.608256E-01 3.657078E-01 2.347905E-01 8.089983E-02 3.609981E-02

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-4.773773E-02 -1.385444E-01 -2.157940E-01 -2.719202E-01 -3.014276E-01 -3.014276E-01 -2.719202E-01 -2.157940E-01 -1.385444E-01 -4.773773E-02 -4.306890E-02 -1.249954E-01 -1.946914E-01 -2.453294E-01 -2.719514E-01 -2.719514E-01 -2.453293E-01 -1.946913E-01 -1.249954E-01 -4.306890E-02 -3.418072E-02 -9.919986E-02 -1.545126E-01 -1.947004E-01 -2.158285E-01 -2.158285E-01 -1.947004E-01 -1.545126E-01 -9.919986E-02 -3.418072E-02 -2.194549E-02 -6.369055E-02 -9.920368E-02 -1.250060E-01 -1.385711E-01 -1.385711E-01 -1.250060E-01 -9.920368E-02 -6.369055E-02 -2.194549E-02 -7.561900E-03 0.0 -2.194626E-02 -3.418324E-02 -4.307410E-02 -4.774832E-02 -4.774832E-02 -4.307409E-02 -3.418323E-02 -2.194626E-02 -7.561900E-03 7.561900E-03 2.194626E-02 3.418324E-02 4.307410E-02 4.774832E-02 4.774832E-02 4.307409E-02 3.418323E-02 2.194626E-02 7.561900E-03 2.194549E-02 6.369055E-02 9.920368E-02 1.250060E-01 1.385711E-01 1.385711E-01 1.250060E-01 9.920368E-02 6.369055E-02 2.194549E-02 3.418072E-02 9.919986E-02 1.545126E-01 1.947004E-01 2.158285E-01 2.158285E-01 1.947004E-01 1.545126E-01 9.919986E-02 3.418072E-02 4.306890E-02 1.249954E-01 1.946914E-01 2.453294E-01 2.719514E-01 2.719514E-01 2.453293E-01 1.946913E-01 1.249954E-01 4.306890E-02 4.773773E-02

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

7-117

Flutter Analysis Sample Problems

Chapter 7

10002 G 0.0 10003 G 0.0 10004 G 0.0 10005 G 0.0 10006 G 0.0 10007 G 0.0 10008 G 0.0 10009 G 0.0 10010 G 0.0 EIGENVALUE = 1.585938E+06 CYCLES = 2.004302E+02 POINT ID.

7-118

TYPE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R E A L T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

1.047696E-01 1.631877E-01 2.056314E-01 2.279455E-01 2.279455E-01 2.056314E-01 1.631877E-01 1.047696E-01 3.609981E-02

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

E I G E N V E C T O R T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.356649E-01 4.273136E-01 5.435858E-01 5.732042E-01 5.260984E-01 4.275815E-01 3.079631E-01 1.916906E-01 9.004164E-02 0.0 0.0 4.018800E-01 7.246771E-01 9.128082E-01 9.480194E-01 8.512367E-01 6.712680E-01 4.650142E-01 2.768810E-01 1.251235E-01 0.0 0.0 4.538650E-01 8.087885E-01 9.969181E-01 1.000000E+00 8.512324E-01 6.192792E-01 3.808975E-01 1.927649E-01 7.313786E-02 0.0 0.0 3.865780E-01 6.714876E-01 7.877569E-01 7.241048E-01 5.260893E-01 2.766623E-01 6.377798E-02 -5.249493E-02 -6.087280E-02 0.0 0.0 2.352183E-01 3.805875E-01 3.805856E-01 2.352143E-01 4.570274E-14 -2.352143E-01 -3.805856E-01 -3.805875E-01 -2.352183E-01 0.0 0.0 6.087280E-02 5.249493E-02 -6.377798E-02 -2.766623E-01 -5.260893E-01 -7.241048E-01 -7.877569E-01 -6.714876E-01 -3.865780E-01 0.0 0.0

1.385444E-01 2.157940E-01 2.719202E-01 3.014276E-01 3.014276E-01 2.719202E-01 2.157940E-01 1.385444E-01 4.773773E-02 N O .

R1 -2.427032E-03 -2.042162E-03 -9.125334E-04 4.719342E-04 1.637925E-03 2.144940E-03 1.831866E-03 8.544188E-04 -4.132396E-04 -1.425787E-03 -1.861258E-03 2.407061E-01 2.176460E-01 1.558882E-01 7.227418E-02 -1.161800E-02 -7.671157E-02 -1.125037E-01 -1.196851E-01 -1.084777E-01 -9.352356E-02 -8.728099E-02 4.129420E-01 3.711607E-01 2.595806E-01 1.098559E-01 -3.757392E-02 -1.474027E-01 -2.009295E-01 -2.009573E-01 -1.684836E-01 -1.326589E-01 -1.181246E-01 4.689076E-01 4.164450E-01 2.768967E-01 9.260224E-02 -8.280083E-02 -2.033237E-01 -2.461843E-01 -2.182660E-01 -1.512395E-01 -8.746059E-02 -6.224748E-02 4.033224E-01 3.492150E-01 2.062137E-01 2.216039E-02 -1.430006E-01 -2.391683E-01 -2.439832E-01 -1.699778E-01 -5.840428E-02 3.776640E-02 7.503222E-02 2.515194E-01 2.034765E-01 7.771879E-02 -7.772297E-02 -2.034771E-01 -2.515111E-01 -2.034771E-01 -7.772297E-02 7.771879E-02 2.034765E-01 2.515194E-01 7.503222E-02 3.776640E-02 -5.840428E-02 -1.699778E-01 -2.439832E-01 -2.391683E-01 -1.430006E-01 2.216039E-02 2.062137E-01 3.492150E-01 4.033224E-01 -6.224748E-02

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

2 R2 4.922156E-03 -2.448997E-01 -4.484198E-01 -5.735155E-01 -6.083425E-01 -5.625579E-01 -4.615641E-01 -3.360363E-01 -2.109802E-01 -9.825688E-02 4.669199E-03 4.016065E-03 -2.083704E-01 -3.793500E-01 -4.805576E-01 -5.023930E-01 -4.551033E-01 -3.631445E-01 -2.552617E-01 -1.540925E-01 -6.924665E-02 3.740484E-03 1.594255E-03 -1.121190E-01 -1.975692E-01 -2.362347E-01 -2.244534E-01 -1.738288E-01 -1.061428E-01 -4.481520E-02 -6.181870E-03 6.078871E-03 1.371021E-03 -1.397782E-03 1.010814E-02 3.239833E-02 7.103771E-02 1.223546E-01 1.738382E-01 2.082595E-01 2.100268E-01 1.713627E-01 9.593305E-02 -1.568791E-03 -3.836978E-03 1.162547E-01 2.302049E-01 3.313861E-01 4.101968E-01 4.551048E-01 4.553460E-01 4.044348E-01 3.032410E-01 1.613606E-01 -3.923689E-03 -4.797455E-03 1.715761E-01 3.296865E-01 4.547616E-01 5.349331E-01 5.625395E-01 5.349331E-01 4.547616E-01 3.296865E-01 1.715761E-01 -4.797455E-03 -3.923689E-03 1.613606E-01 3.032410E-01 4.044348E-01 4.553460E-01 4.551048E-01 4.101968E-01 3.313861E-01 2.302049E-01 1.162547E-01 -3.836978E-03 -1.568791E-03

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

85

79 80 81 82 83 84 G 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 3001 3002 3003 3004 3005 3006 3007 3008 3009 3010 4001 4002 4003 4004 4005 4006 4007 4008 4009 4010

5001

G G G G G G 0.0 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

5002 5003 5004 5005 5006 5007 5008

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

-7.313786E-02 -8.746059E-02 9.593305E-02 -1.927649E-01 -1.512395E-01 1.713627E-01 -3.808975E-01 -2.182660E-01 2.100268E-01 -6.192792E-01 -2.461843E-01 2.082595E-01 -8.512324E-01 -2.033237E-01 1.738382E-01 -1.000000E+00 -8.280083E-02 1.223546E-01 -9.969181E-01 9.260224E-02 7.103771E-02 0.0 -8.087885E-01 2.768967E-01 3.239833E-02 -4.538650E-01 4.164450E-01 1.010814E-02 0.0 4.689076E-01 -1.397782E-03 0.0 -1.181246E-01 1.371021E-03 -1.251235E-01 -1.326589E-01 6.078871E-03 -2.768810E-01 -1.684836E-01 -6.181870E-03 -4.650142E-01 -2.009573E-01 -4.481520E-02 -6.712680E-01 -2.009295E-01 -1.061428E-01 -8.512367E-01 -1.474027E-01 -1.738288E-01 -9.480194E-01 -3.757392E-02 -2.244534E-01 -9.128082E-01 1.098559E-01 -2.362347E-01 -7.246771E-01 2.595806E-01 -1.975692E-01 -4.018800E-01 3.711607E-01 -1.121190E-01 0.0 4.129420E-01 1.594255E-03 0.0 -8.728099E-02 3.740484E-03 -9.004164E-02 -9.352356E-02 -6.924665E-02 -1.916906E-01 -1.084777E-01 -1.540925E-01 -3.079631E-01 -1.196851E-01 -2.552617E-01 -4.275815E-01 -1.125037E-01 -3.631445E-01 -5.260984E-01 -7.671157E-02 -4.551033E-01 -5.732042E-01 -1.161800E-02 -5.023930E-01 -5.435858E-01 7.227418E-02 -4.805576E-01 -4.273136E-01 1.558882E-01 -3.793500E-01 -2.356649E-01 2.176460E-01 -2.083704E-01 0.0 2.407061E-01 4.016065E-03 0.0 -1.861258E-03 4.669199E-03 0.0 -1.425787E-03 -9.825688E-02 0.0 -4.132396E-04 -2.109802E-01 0.0 8.544188E-04 -3.360363E-01 0.0 1.831866E-03 -4.615641E-01 0.0 2.144940E-03 -5.625579E-01 0.0 1.637925E-03 -6.083425E-01 0.0 4.719342E-04 -5.735155E-01 0.0 -9.125334E-04 -4.484198E-01 0.0 -2.042162E-03 -2.448997E-01 0.0 -2.427032E-03 4.922156E-03 3.090505E-02 0.0 -1.178324E-01 8.678451E-02 0.0 -3.314892E-01 1.266528E-01 0.0 -4.854497E-01 1.449058E-01 0.0 -5.583950E-01 1.415278E-01 0.0 -5.496513E-01 1.214685E-01 0.0 -4.768399E-01 9.238999E-02 0.0 -3.677723E-01 6.173965E-02 0.0 -2.498269E-01 3.426860E-02 0.0 -1.408661E-01 1.084573E-02 0.0 -4.502082E-02 1.426601E-01 0.0 -8.310755E-02 4.003652E-01 0.0 -2.317893E-01 5.834715E-01 0.0 -3.332930E-01 6.659800E-01 0.0 -3.720188E-01 6.481806E-01 0.0 -3.499767E-01 5.535938E-01 0.0 -2.844124E-01 4.183482E-01 0.0 -2.003687E-01 2.773813E-01 0.0 -1.211207E-01 1.527319E-01 0.0 -6.013613E-02 4.807205E-02 0.0 -1.754093E-02 2.133086E-01 0.0 -2.599249E-02 5.960586E-01 0.0 -6.804819E-02 8.607312E-01 0.0 -8.411065E-02 9.677065E-01 0.0 -6.804526E-02 9.206281E-01 0.0 -2.598815E-02 7.608748E-01 0.0 2.599661E-02 5.494665E-01 0.0 6.805279E-02 3.437566E-01 0.0 8.411642E-02 1.778140E-01 0.0 6.805089E-02 5.355187E-02 0.0 2.599282E-02 2.251595E-01 0.0 3.364347E-02 6.234537E-01 0.0 1.022939E-01 8.826140E-01 0.0 1.732311E-01 9.591787E-01 0.0 2.425282E-01 8.641079E-01 0.0 3.005191E-01 6.548499E-01 0.0 3.338800E-01 4.113334E-01 0.0 3.298682E-01 2.054378E-01 0.0 2.811897E-01 7.560256E-02 0.0 1.896352E-01 1.589719E-02 0.0 6.700533E-02 1.807391E-01 0.0 7.567985E-02 0.0 4.905497E-01 0.0 2.211299E-01 6.635824E-01 0.0 3.490358E-01 6.621167E-01 0.0 4.480309E-01 5.072617E-01 0.0 5.074899E-01 2.689662E-01 0.0 5.189829E-01 3.825030E-02 0.0 4.781200E-01 -1.075615E-01 0.0 3.862281E-01

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

7-119

Flutter Analysis Sample Problems

Chapter 7

0

5009 5010 6001 6002 6003 6004 6005 6006 6007 6008 6009 6010 7001 7002 7003 7004 7005 7006 7007 7008 7009 7010 8001 8002 8003 8004 8005 8006 8007 8008 8009 8010 9001 9002 9003 9004 9005 9006 9007 9008 9009 9010 10001 10002 10003 10004 10005 10006 10007 10008 10009 10010 EIGENVALUE = CYCLES =

POINT ID.

7-120

TYPE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 1.585938E+06 2.004302E+02 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R E A L T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

-1.333238E-01 -5.756161E-02 1.011480E-01 2.589333E-01 3.006755E-01 2.008096E-01 -9.474851E-03 -2.535168E-01 -4.381014E-01 -4.890645E-01 -3.799848E-01 -1.428992E-01 1.760547E-02 1.921505E-02 -6.484304E-02 -2.463995E-01 -4.879100E-01 -7.138484E-01 -8.379146E-01 -7.959984E-01 -5.723068E-01 -2.083378E-01 -4.055545E-02 -1.437885E-01 -3.016984E-01 -5.154403E-01 -7.478766E-01 -9.336222E-01 -1.001729E+00 -9.027865E-01 -6.300827E-01 -2.263049E-01 -5.684251E-02 -1.828000E-01 -3.379417E-01 -5.185327E-01 -6.958001E-01 -8.231691E-01 -8.519893E-01 -7.501180E-01 -5.162599E-01 -1.842139E-01 -3.335614E-02 -1.047017E-01 -1.866531E-01 -2.762762E-01 -3.598885E-01 -4.163534E-01 -4.241033E-01 -3.693776E-01 -2.525291E-01 -8.982126E-02

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

E I G E N V E C T O R T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -9.004164E-02 -1.251235E-01 -7.313786E-02 6.087280E-02 2.352183E-01 3.865780E-01 4.538650E-01 4.018800E-01 2.356649E-01 0.0 0.0 -1.916906E-01 -2.768810E-01 -1.927649E-01 5.249493E-02 3.805875E-01 6.714876E-01 8.087885E-01 7.246771E-01 4.273136E-01 0.0 0.0 -3.079631E-01

2.512190E-01 8.717276E-02 8.717276E-02 2.512190E-01 3.862281E-01 4.781201E-01 5.189829E-01 5.074899E-01 4.480309E-01 3.490357E-01 2.211299E-01 7.567985E-02 6.700533E-02 1.896352E-01 2.811897E-01 3.298682E-01 3.338800E-01 3.005191E-01 2.425282E-01 1.732311E-01 1.022939E-01 3.364347E-02 2.599282E-02 6.805089E-02 8.411642E-02 6.805277E-02 2.599659E-02 -2.598817E-02 -6.804528E-02 -8.411065E-02 -6.804819E-02 -2.599249E-02 -1.754093E-02 -6.013613E-02 -1.211208E-01 -2.003688E-01 -2.844124E-01 -3.499768E-01 -3.720188E-01 -3.332930E-01 -2.317893E-01 -8.310755E-02 -4.502082E-02 -1.408661E-01 -2.498269E-01 -3.677724E-01 -4.768400E-01 -5.496513E-01 -5.583950E-01 -4.854497E-01 -3.314892E-01 -1.178324E-01 N O .

R1 4.669199E-03 3.740484E-03 1.371021E-03 -1.568791E-03 -3.923689E-03 -4.797455E-03 -3.836978E-03 -1.397782E-03 1.594255E-03 4.016065E-03 4.922156E-03 -9.825688E-02 -6.924665E-02 6.078871E-03 9.593305E-02 1.613606E-01 1.715761E-01 1.162547E-01 1.010814E-02 -1.121190E-01 -2.083704E-01 -2.448997E-01 -2.109802E-01 -1.540925E-01 -6.181870E-03 1.713627E-01 3.032410E-01 3.296865E-01 2.302049E-01 3.239833E-02 -1.975692E-01 -3.793500E-01 -4.484198E-01 -3.360363E-01 -2.552617E-01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

3 R2 1.861258E-03 8.728099E-02 1.181246E-01 6.224748E-02 -7.503222E-02 -2.515194E-01 -4.033224E-01 -4.689076E-01 -4.129420E-01 -2.407061E-01 2.427032E-03 1.425787E-03 9.352356E-02 1.326589E-01 8.746059E-02 -3.776640E-02 -2.034765E-01 -3.492150E-01 -4.164450E-01 -3.711607E-01 -2.176460E-01 2.042162E-03 4.132396E-04 1.084777E-01 1.684836E-01 1.512395E-01 5.840428E-02 -7.771879E-02 -2.062137E-01 -2.768967E-01 -2.595806E-01 -1.558882E-01 9.125334E-04 -8.544188E-04 1.196851E-01

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

76

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 G 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 1001 1002 1003 1004 1005

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G 0.0 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-4.650142E-01 -4.481520E-02 2.009573E-01 -3.808975E-01 2.100268E-01 2.182660E-01 -6.377798E-02 4.044348E-01 1.699778E-01 3.805856E-01 4.547616E-01 7.772297E-02 7.877569E-01 3.313861E-01 -2.216039E-02 9.969181E-01 7.103771E-02 -9.260224E-02 9.128082E-01 -2.362347E-01 -1.098559E-01 5.435858E-01 -4.805576E-01 -7.227418E-02 0.0 -5.735155E-01 -4.719342E-04 0.0 -4.615641E-01 -1.831866E-03 -4.275815E-01 -3.631445E-01 1.125037E-01 -6.712680E-01 -1.061428E-01 2.009295E-01 -6.192792E-01 2.082595E-01 2.461843E-01 -2.766623E-01 4.553460E-01 2.439832E-01 2.352143E-01 5.349331E-01 2.034771E-01 7.241048E-01 4.101968E-01 1.430006E-01 1.000000E+00 1.223546E-01 8.280083E-02 9.480194E-01 -2.244534E-01 3.757392E-02 5.732042E-01 -5.023930E-01 1.161800E-02 0.0 -6.083425E-01 -1.637925E-03 0.0 -5.625579E-01 -2.144940E-03 -5.260984E-01 -4.551033E-01 7.671157E-02 -8.512367E-01 -1.738288E-01 1.474027E-01 -8.512324E-01 1.738382E-01 2.033237E-01 -5.260893E-01 4.551048E-01 2.391683E-01 1.993383E-13 5.625395E-01 2.515111E-01 5.260893E-01 4.551048E-01 2.391683E-01 8.512324E-01 1.738382E-01 2.033237E-01 8.512367E-01 -1.738288E-01 1.474027E-01 5.260984E-01 -4.551033E-01 7.671157E-02 0.0 -5.625579E-01 -2.144940E-03 0.0 -6.083425E-01 -1.637925E-03 -5.732042E-01 -5.023930E-01 1.161800E-02 -9.480194E-01 -2.244534E-01 3.757392E-02 -1.000000E+00 1.223546E-01 8.280083E-02 -7.241048E-01 4.101968E-01 1.430006E-01 -2.352143E-01 5.349331E-01 2.034771E-01 2.766623E-01 4.553460E-01 2.439832E-01 6.192792E-01 2.082595E-01 2.461843E-01 6.712680E-01 -1.061428E-01 2.009295E-01 4.275815E-01 -3.631445E-01 1.125037E-01 0.0 0.0 -4.615641E-01 -1.831866E-03 0.0 -5.735155E-01 -4.719342E-04 -5.435858E-01 -4.805576E-01 -7.227418E-02 -9.128082E-01 -2.362347E-01 -1.098559E-01 -9.969181E-01 7.103771E-02 -9.260224E-02 -7.877569E-01 3.313861E-01 -2.216039E-02 -3.805856E-01 4.547616E-01 7.772297E-02 6.377798E-02 4.044348E-01 1.699778E-01 3.808975E-01 2.100268E-01 2.182660E-01 4.650142E-01 -4.481520E-02 2.009573E-01 3.079631E-01 -2.552617E-01 1.196851E-01 0.0 -3.360363E-01 -8.544188E-04 0.0 -4.484198E-01 9.125334E-04 -4.273136E-01 -3.793500E-01 -1.558882E-01 -7.246771E-01 -1.975692E-01 -2.595806E-01 -8.087885E-01 3.239833E-02 -2.768967E-01 -6.714876E-01 2.302049E-01 -2.062137E-01 -3.805875E-01 3.296865E-01 -7.771879E-02 -5.249493E-02 3.032410E-01 5.840428E-02 1.927649E-01 1.713627E-01 1.512395E-01 2.768810E-01 -6.181870E-03 1.684836E-01 1.916906E-01 -1.540925E-01 1.084777E-01 0.0 -2.109802E-01 4.132396E-04 0.0 -2.448997E-01 2.042162E-03 -2.356649E-01 -2.083704E-01 -2.176460E-01 -4.018800E-01 -1.121190E-01 -3.711607E-01 -4.538650E-01 1.010814E-02 -4.164450E-01 -3.865780E-01 1.162547E-01 -3.492150E-01 -2.352183E-01 1.715761E-01 -2.034765E-01 -6.087280E-02 1.613606E-01 -3.776640E-02 7.313786E-02 9.593305E-02 8.746059E-02 1.251235E-01 6.078871E-03 1.326589E-01 9.004164E-02 -6.924665E-02 9.352356E-02 0.0 -9.825688E-02 1.425787E-03 0.0 4.922156E-03 2.427032E-03 0.0 4.016065E-03 -2.407061E-01 0.0 1.594255E-03 -4.129420E-01 0.0 -1.397782E-03 -4.689076E-01 0.0 -3.836978E-03 -4.033224E-01 0.0 -4.797455E-03 -2.515194E-01 0.0 -3.923689E-03 -7.503222E-02 0.0 -1.568791E-03 6.224748E-02 0.0 1.371021E-03 1.181246E-01 0.0 3.740484E-03 8.728099E-02 0.0 4.669199E-03 1.861258E-03 -1.309935E-02 0.0 4.502082E-02 -3.166817E-02 0.0 1.075826E-01 -3.047854E-02 0.0 9.913067E-02 -5.681969E-03 0.0 6.132499E-03 3.636016E-02 0.0 -1.480456E-01

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

7-121

Flutter Analysis Sample Problems

Chapter 7

1006 1007 1008 1009 1010 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 3001 3002 3003 3004 3005 3006 3007 3008 3009 3010 4001

G 4002 4003 4004 4005 4006 4007 4008 4009 4010 5001 5002 5003 5004 5005 5006 5007 5008 5009 5010 6001 6002 6003 6004 6005 6006 6007 6008 6009 6010 7001 7002 7003 7004 7005 7006 7007 7008 7009 7010 8001 8002 8003 8004 8005 8006 8007 8008 8009 8010 9001 9002 9003 9004 9005 9006 9007 9008 9009 9010 10001 10002 10003 10004 10005 10006

7-122

G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

8.122538E-02 1.117768E-01 1.147087E-01 8.579095E-02 3.176610E-02 -6.314153E-02 -1.532884E-01 -1.498631E-01 -3.452995E-02 1.628782E-01 3.744347E-01 5.193326E-01 5.347499E-01 4.005778E-01 1.484178E-01 -1.190025E-01 -2.947320E-01 -3.088797E-01 -1.285621E-01 1.972445E-01 5.545548E-01 8.068895E-01 8.474056E-01 6.407301E-01 2.383157E-01 -1.801340E-01 0.0 -4.563987E-01 -5.140654E-01 -3.062881E-01 1.055455E-01 5.735800E-01 9.190436E-01 1.000533E+00 7.689465E-01 2.879103E-01 -2.389600E-01 -6.178321E-01 -7.380867E-01 -5.411148E-01 -9.296473E-02 4.400161E-01 8.542488E-01 9.866834E-01 7.775476E-01 2.940325E-01 -2.822561E-01 -7.415755E-01 -9.252958E-01 -7.675529E-01 -3.317084E-01 2.141253E-01 6.614601E-01 8.410672E-01 6.874534E-01 2.635892E-01 -2.953149E-01 -7.851539E-01 -1.010106E+00 -9.039009E-01 -5.213240E-01 -1.598135E-02 4.191048E-01 6.252244E-01 5.378668E-01 2.100386E-01 -2.673837E-01 -7.168308E-01 -9.414708E-01 -8.829891E-01 -5.836215E-01 -1.681756E-01 2.046636E-01 4.029462E-01 3.708335E-01 1.480706E-01 -1.963300E-01 -5.291893E-01 -7.041802E-01 -6.792909E-01 -4.820043E-01 -1.971259E-01 6.653132E-02 2.177092E-01 2.166400E-01 8.855377E-02 -9.068232E-02 -2.451772E-01 -3.286449E-01 -3.218875E-01 -2.366744E-01 -1.103829E-01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-3.108982E-01 -4.202215E-01 -4.278724E-01 -3.187724E-01 -1.178324E-01 5.082448E-02 1.267032E-01 1.356923E-01 6.400242E-02 -6.849566E-02 -2.151395E-01 -3.199167E-01 -3.388604E-01 -2.572230E-01 -9.582436E-02 5.813625E-02 1.522028E-01 1.881329E-01 1.522027E-01 5.813742E-02 -5.813370E-02 -1.521994E-01 -1.881303E-01 -1.522016E-01 -5.813611E-02 5.980922E-02 0.0 1.629362E-01 2.223178E-01 2.256330E-01 1.791278E-01 1.045117E-01 3.028506E-02 -1.914653E-02 -3.241478E-02 -1.480919E-02 4.925841E-02 1.392427E-01 2.059609E-01 2.406901E-01 2.423206E-01 2.166149E-01 1.733915E-01 1.227751E-01 7.194424E-02 2.355291E-02 2.355291E-02 7.194424E-02 1.227752E-01 1.733916E-01 2.166149E-01 2.423206E-01 2.406901E-01 2.059609E-01 1.392427E-01 4.925841E-02 -1.480919E-02 -3.241478E-02 -1.914652E-02 3.028511E-02 1.045117E-01 1.791279E-01 2.256331E-01 2.223178E-01 1.629362E-01 5.980922E-02 -5.813611E-02 -1.522016E-01 -1.881303E-01 -1.521994E-01 -5.813365E-02 5.813748E-02 1.522028E-01 1.881329E-01 1.522028E-01 5.813625E-02 -9.582436E-02 -2.572230E-01 -3.388604E-01 -3.199166E-01 -2.151394E-01 -6.849559E-02 6.400249E-02 1.356923E-01 1.267032E-01 5.082448E-02 -1.178324E-01 -3.187724E-01 -4.278725E-01 -4.202214E-01 -3.108982E-01 -1.480455E-01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

FLUTTER POINT = KFREQ 0.2087 0.2347 0.2323 0.2302 0.2243 0.2218 0.2216 0.2216 0.2220 POINT = KFREQ 0.2722 0.2348 0.2323 0.2302 0.2243 0.2218 0.2216 0.2216 0.2220 POINT = KFREQ 0.3140 0.3027 0.2978 0.2933 0.2799 0.2732 0.2726 0.2726 0.2746 POINT = KFREQ 0.4678 0.4432 0.4318 0.4210 0.3823 0.3489 0.3410 0.3332 0.3169 POINT = KFREQ 0.1505 0.1504 0.1508 0.1514 0.1575 0.1763 0.1831 0.1817 0.1793 POINT = KFREQ 0.2876 0.2712 0.2635 0.2561 0.2278 0.1932 0.1832 0.1817 0.1793 POINT = KFREQ 0.3000 0.2861 0.2799 0.2740 0.2540 0.2388 0.2356 0.2327 0.2275 POINT = KFREQ 0.4675 0.4430 0.4317 0.4209 0.3828 0.3510 0.3438 0.3369 0.3239

10007 G 0.0 0.0 8.748291E-03 0.0 6.132595E-03 0.0 10008 G 0.0 0.0 8.004388E-02 0.0 9.913068E-02 0.0 10009 G 0.0 0.0 8.545946E-02 0.0 1.075826E-01 0.0 10010 G 0.0 0.0 3.560976E-02 0.0 4.502082E-02 0.0 SUMMARY 1 MACH NUMBER = 2.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 4.7904491E+00 1.8000000E+03 -1.5183085E-01 1.4352501E+02 -6.8460098E+01 9.0179431E+02 4.2612085E+00 1.9000000E+03 -2.2514819E-01 1.7031471E+02 -1.2046767E+02 1.0701189E+03 4.3047180E+00 1.9500000E+03 -3.4828857E-01 1.7302994E+02 -1.8932603E+02 1.0871792E+03 4.3440037E+00 2.0000000E+03 -4.2062744E-01 1.7586166E+02 -2.3239066E+02 1.1049714E+03 4.4580016E+00 2.2000000E+03 -5.9151113E-01 1.8850107E+02 -3.5028909E+02 1.1843872E+03 4.5085921E+00 2.4000000E+03 -6.8576783E-01 2.0333009E+02 -4.3805499E+02 1.2775607E+03 4.5123258E+00 2.4500000E+03 -7.0259273E-01 2.0739436E+02 -4.5777335E+02 1.3030973E+03 4.5127783E+00 2.5000000E+03 -7.1737355E-01 2.1160571E+02 -4.7689487E+02 1.3295580E+03 4.5044394E+00 2.6000000E+03 -7.4176371E-01 2.2047734E+02 -5.1378265E+02 1.3853000E+03 2 MACH NUMBER = 2.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 3.6741130E+00 1.8000000E+03 -1.1624902E-01 1.8713342E+02 -6.8342453E+01 1.1757941E+03 4.2597957E+00 1.9000000E+03 -4.4721335E-02 1.7037120E+02 -2.3936510E+01 1.0704739E+03 4.3041105E+00 1.9500000E+03 7.5636230E-02 1.7305435E+02 4.1120872E+01 1.0873326E+03 4.3435454E+00 2.0000000E+03 1.4548054E-01 1.7588022E+02 8.0384399E+01 1.1050880E+03 4.4577065E+00 2.2000000E+03 3.0912995E-01 1.8851357E+02 1.8307692E+02 1.1844657E+03 4.5083399E+00 2.4000000E+03 4.0016800E-01 2.0334145E+02 2.5563374E+02 1.2776321E+03 4.5120788E+00 2.4500000E+03 4.1675222E-01 2.0740573E+02 2.7154922E+02 1.3031687E+03 4.5125346E+00 2.5000000E+03 4.3149984E-01 2.1161714E+02 2.8686752E+02 1.3296298E+03 4.5041981E+00 2.6000000E+03 4.5640814E-01 2.2048915E+02 3.1614804E+02 1.3853743E+03 3 MACH NUMBER = 2.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 3.1845033E+00 1.8000000E+03 -1.0585071E-01 2.1590474E+02 -7.1796928E+01 1.3565696E+03 3.3038633E+00 1.9000000E+03 -1.0982659E-01 2.1966603E+02 -7.5791466E+01 1.3802024E+03 3.3583839E+00 1.9500000E+03 -1.1164407E-01 2.2178677E+02 -7.7789536E+01 1.3935275E+03 3.4092307E+00 2.0000000E+03 -1.1334021E-01 2.2408096E+02 -7.9788239E+01 1.4079423E+03 3.5722883E+00 2.2000000E+03 -1.1879526E-01 2.3523805E+02 -8.7792328E+01 1.4780443E+03 3.6603711E+00 2.4000000E+03 -1.2179092E-01 2.5044794E+02 -9.5825752E+01 1.5736108E+03 3.6681280E+00 2.4500000E+03 -1.2207674E-01 2.5512497E+02 -9.7844360E+01 1.6029976E+03 3.6687770E+00 2.5000000E+03 -1.2213536E-01 2.6028555E+02 -9.9871452E+01 1.6354225E+03 3.6413260E+00 2.6000000E+03 -1.2136049E-01 2.7273770E+02 -1.0398540E+02 1.7136615E+03 4 MACH NUMBER = 2.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 2.1377494E+00 1.8000000E+03 -7.0856951E-02 3.2162299E+02 -7.1594460E+01 2.0208169E+03 2.2563396E+00 1.9000000E+03 -7.4784920E-02 3.2164771E+02 -7.5569122E+01 2.0209723E+03 2.3157427E+00 1.9500000E+03 -7.6752007E-02 3.2164417E+02 -7.7555984E+01 2.0209500E+03 2.3752625E+00 2.0000000E+03 -7.8722581E-02 3.2162494E+02 -7.9542442E+01 2.0208291E+03 2.6157110E+00 2.2000000E+03 -8.6677067E-02 3.2126566E+02 -8.7481941E+01 2.0185718E+03 2.8661566E+00 2.4000000E+03 -9.4939411E-02 3.1984729E+02 -9.5397964E+01 2.0096599E+03 2.9324751E+00 2.4500000E+03 -9.7118646E-02 3.1912668E+02 -9.7367859E+01 2.0051321E+03 3.0016396E+00 2.5000000E+03 -9.9384069E-02 3.1813602E+02 -9.9329788E+01 1.9989076E+03 3.1559603E+00 2.6000000E+03 -1.0438508E-01 3.1468289E+02 -1.0319566E+02 1.9772111E+03 5 MACH NUMBER = 3.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 6.6425405E+00 1.8000000E+03 -1.2886114E-01 1.0350698E+02 -4.1902653E+01 6.5035358E+02 6.6493044E+00 1.9000000E+03 -1.2899652E-01 1.0914623E+02 -4.4232006E+01 6.8578601E+02 6.6328869E+00 1.9500000E+03 -1.2868065E-01 1.1229577E+02 -4.5396935E+01 7.0557513E+02 6.6033998E+00 2.0000000E+03 -1.2811172E-01 1.1568945E+02 -4.6562096E+01 7.2689832E+02 6.3485928E+00 2.2000000E+03 -1.2318996E-01 1.3236604E+02 -5.1227341E+01 8.3168042E+02 5.6734242E+00 2.4000000E+03 -1.1028760E-01 1.6158362E+02 -5.5985287E+01 1.0152599E+03 5.4600530E+00 2.4500000E+03 -2.5049758E-01 1.7139595E+02 -1.3488200E+02 1.0769126E+03 5.5026503E+00 2.5000000E+03 -3.2879058E-01 1.7353995E+02 -1.7925394E+02 1.0903837E+03 5.5774884E+00 2.6000000E+03 -4.2926401E-01 1.7805984E+02 -2.4012665E+02 1.1187831E+03 6 MACH NUMBER = 3.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 3.4776528E+00 1.8000000E+03 -6.7410626E-02 1.9770499E+02 -4.1869316E+01 1.2422172E+03 3.6873910E+00 1.9000000E+03 -7.1474388E-02 1.9681844E+02 -4.4194286E+01 1.2366467E+03 3.7951260E+00 1.9500000E+03 -7.3561460E-02 1.9626361E+02 -4.5356548E+01 1.2331606E+03 3.9053950E+00 2.0000000E+03 -7.5697303E-02 1.9561240E+02 -4.6518604E+01 1.2290690E+03 4.3902888E+00 2.2000000E+03 -8.5082710E-02 1.9140839E+02 -5.1162548E+01 1.2026544E+03 5.1759119E+00 2.4000000E+03 -1.0012950E-01 1.7711516E+02 -5.5714428E+01 1.1128474E+03 5.4591718E+00 2.4500000E+03 3.8724344E-02 1.7142361E+02 2.0854733E+01 1.0770864E+03 5.5020719E+00 2.5000000E+03 1.1535835E-01 1.7355818E+02 6.2899044E+01 1.0904983E+03 5.5770836E+00 2.6000000E+03 2.1292362E-01 1.7807277E+02 1.1911631E+02 1.1188643E+03 7 MACH NUMBER = 3.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 3.3334911E+00 1.8000000E+03 -6.7840591E-02 2.0625502E+02 -4.3958622E+01 1.2959386E+03 3.4946914E+00 1.9000000E+03 -7.1122944E-02 2.0767114E+02 -4.6401901E+01 1.3048363E+03 3.5729437E+00 1.9500000E+03 -7.2716504E-02 2.0846820E+02 -4.7623653E+01 1.3098444E+03 3.6494915E+00 2.0000000E+03 -7.4275538E-02 2.0932880E+02 -4.8845512E+01 1.3152517E+03 3.9365015E+00 2.2000000E+03 -8.0122933E-02 2.1347331E+02 -5.3734142E+01 1.3412925E+03 4.1876168E+00 2.4000000E+03 -8.5243277E-02 2.1891507E+02 -5.8625381E+01 1.3754840E+03 4.2439756E+00 2.4500000E+03 -8.6393394E-02 2.2050812E+02 -5.9848740E+01 1.3854934E+03 4.2975521E+00 2.5000000E+03 -8.7487258E-02 2.2220316E+02 -6.1072392E+01 1.3961437E+03 4.3959293E+00 2.6000000E+03 -8.9497574E-02 2.2591963E+02 -6.3520679E+01 1.4194950E+03 8 MACH NUMBER = 3.0000 DENSITY RATIO = 5.0000E-01 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 2.1388676E+00 1.8000000E+03 -4.3416854E-02 3.2145486E+02 -4.3845825E+01 2.0197605E+03 2.2573581E+00 1.9000000E+03 -4.5821652E-02 3.2150262E+02 -4.6281258E+01 2.0200607E+03 2.3165796E+00 1.9500000E+03 -4.7023520E-02 3.2152795E+02 -4.7498924E+01 2.0202198E+03 2.3757868E+00 2.0000000E+03 -4.8225053E-02 3.2155399E+02 -4.8716545E+01 2.0203833E+03 2.6125252E+00 2.2000000E+03 -5.3028762E-02 3.2165741E+02 -5.3586445E+01 2.0210332E+03 2.8493795E+00 2.4000000E+03 -5.7833485E-02 3.2173053E+02 -5.8454983E+01 2.0214927E+03 2.9086857E+00 2.4500000E+03 -5.9036233E-02 3.2173672E+02 -5.9671806E+01 2.0215316E+03 2.9680619E+00 2.5000000E+03 -6.0240243E-02 3.2173508E+02 -6.0888466E+01 2.0215212E+03 3.0871236E+00 2.6000000E+03 -6.2653899E-02 3.2169971E+02 -6.3321129E+01 2.0212990E+03

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Chapter 7

7-124

Flutter Analysis Sample Problems

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

Listing 7-32. Output for Flutter Analysis of Square Panel Using Piston Theory N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA145HB TIME 45 SOL 145 CEND 3D PANEL FLUTTER PAGE 2 PINNED EDGES ZONA51 AERO, EXAMPLE HA145HB C A S E C O N T R O L D E C K E C H O CARD COUNT 1 $ 2 TITLE = 3D PANEL FLUTTER 3 SUBTI = PINNED EDGES 4 LABEL = ZONA51 AERO, EXAMPLE HA145HB 5 ECHO = BOTH 6 METHOD = 20 7 FMETHOD = 30 8 SPC = 1 9 SET 10 = 1 THRU 999999 10 DISP = 10 11 BEGIN BULK 3D PANEL FLUTTER PINNED EDGES ZONA51 AERO, EXAMPLE HA145HB I N P U T B U L K D A T A

PAGE

3

D E C K

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 $ INCLUDE PANEL_STRUCT.DAT $ $ CAERO FOR PANEL SURFACE $ EID PID CP NSPAN NCHORD LSPAN CAERO1 1001 1 0 10 10 $ X1 Y1 Z1 X12 X4 Y4 +CA1 0.0 0.0 0.0 10.0 0.0 10.0 $ CAERO FOR SURFACE INBOARD OF PANEL CAERO1 2001 1 0 10 10 +CA2 0.0 -10.0 0.0 10.0 0.0 0.0 $ CAERO FOR SURFACE OUTBOARD OF SURFACE CAERO1 3001 1 0 10 10 +CA3 0.0 10.0 0.0 10.0 0.0 20.0 $ PAERO1 1 $ $ EID CAERO ID1 ID2 SETG DZ SPLINE2 1001 1001 1001 1091 1001 0.0 $+SPL DTHX DTHY +SPL1 -1.0 -1.0 $ SPLINE2 2001 1001 1002 1092 2001 0.0 +SPL2 -1.0 -1.0 SPLINE2 3001 1001 1003 1093 3001 0.0 +SPL3 -1.0 -1.0 SPLINE2 4001 1001 1004 1094 4001 0.0 +SPL4 -1.0 -1.0 SPLINE2 5001 1001 1005 1095 5001 0.0 +SPL5 -1.0 -1.0 SPLINE2 6001 1001 1006 1096 6001 0.0 +SPL6 -1.0 -1.0 SPLINE2 7001 1001 1007 1097 7001 0.0 +SPL7 -1.0 -1.0 SPLINE2 8001 1001 1008 1098 8001 0.0 +SPL8 -1.0 -1.0 SPLINE2 9001 1001 1009 1099 9001 0.0 +SPL9 -1.0 -1.0 SPLINE2 10001 1001 1010 1100 10001 0.0 +SPL10 -1.0 -1.0

E C H O ..

8

LCHORD

..

9

..

10

IGID 1 X14 10.0

+CA +CA1

Z4 0.0

1 10.0

+CA2

0.0

1 10.0

+CA3

0.0

DTOR 1.0

CID 1

+SPL +SPL1

1.0

2

+SPL2

1.0

3

+SPL3

1.0

4

+SPL4

1.0

5

+SPL5

1.0

6

+SPL6

1.0

7

+SPL7

1.0

8

+SPL8

1.0

9

+SPL9

1.0

10

+SPL10

$ $ SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 $ $ CORD2R $ +CRD1 CORD2R

SID 1001 2001 3001 4001 5001 6001 7001 8001 9001 10001

G1 1 12 23 34 45 56 67 78 89 100

G2 THRU THRU THRU THRU THRU THRU THRU THRU THRU THRU

G3 22 33 44 55 66 77 88 99 110 121

ETC

CID 1 C1 20.0 2

RID

A1 0.5 C3 0.0 1.5

A2 0.0

A3 0.0

B1 0.5

B2 0.0

B3 1.0

+CRD +CRD1

C2 0.0

0.0

0.0

1.5

0.0

1.0

+CRD2

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.

Chapter 7

Flutter Analysis Sample Problems

+CRD2 20.0 CORD2R 3 +CRD3 20.0 CORD2R 4 +CRD4 20.0 CORD2R 5 +CRD5 20.0 CORD2R 6 +CRD6 20.0 CORD2R 7 +CRD7 20.0 CORD2R 8 +CRD8 20.0 CORD2R 9 +CRD9 20.0 CORD2R 10 +CRD10 20.0 $ AERO 0 $ $ M1 MKAERO1 2.0 $ K1 +MK 0.001 $ FLUTTER 30 FLFACT 1 FLFACT 2 FLFACT 3 +FL3 30000. $ PARAM VREF $ EIGR 20 +ER MAX $ $PARAM LMODES $ PARAM OPPHIPA $ ENDDATA INPUT BULK DATA

0.0

0.0 2.5 0.0 3.5 0.0 4.5 0.0 5.5 0.0 6.5 0.0 7.5 0.0 8.5 0.0 9.5 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0

0.0

2.5

0.0

1.0

+CRD3

0.0

0.0

3.5

0.0

1.0

+CRD4

0.0

0.0

4.5

0.0

1.0

+CRD5

0.0

0.0

5.5

0.0

1.0

+CRD6

0.0

0.0

6.5

0.0

1.0

+CRD7

0.0

0.0

7.5

0.0

1.0

+CRD8

0.0

0.0

8.5

0.0

1.0

+CRD9

0.0

0.0

9.5

0.0

1.0

+CRD10

0

10.0

M2 3.0 K2 0.1

ETC

1.1468-7

K3 0.2

ETC 0.4

PK 0.5 2.0 21600. 31200.

1

2

3

3.0 22200.

22800.

24000.

0.

1000.

+MK +MK

4 26400.

28800.

29400.

+FL3

12.0 AGIV

15

+ER

10 1 CARD COUNT =

191

Listing 7-33. Input Data for Flutter Analysis of Square Panel Using ZONA51 Aerodynamics 3D PANEL FLUTTER PINNED EDGES ZONA51 AERO, EXAMPLE HA145HB

PAGE S O R T E D

CARD COUNT 1234567891011121314151617181920212223242526272829303132333435363738-

7-126

. 1 .. 2 AERO 0 CAERO1 1001 +CA1 0.0 CAERO1 2001 +CA2 0.0 CAERO1 3001 +CA3 0.0 CORD2R 1 +CRD1 20.0 CORD2R 2 +CRD2 20.0 CORD2R 3 +CRD3 20.0 CORD2R 4 +CRD4 20.0 CORD2R 5 +CRD5 20.0 CORD2R 6 +CRD6 20.0 CORD2R 7 +CRD7 20.0 CORD2R 8 +CRD8 20.0 CORD2R 9 +CRD9 20.0 CORD2R 10 +CRD10 20.0 CQUAD4 1 CQUAD4 2 CQUAD4 3 CQUAD4 4 CQUAD4 5 CQUAD4 6 CQUAD4 7 CQUAD4 8 CQUAD4 9 CQUAD4 10 CQUAD4 11

NX Nastran Aeroelastic Analysis User’s Guide

.. 3 0 1 0.0 1 -10.0 1 10.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1 1 1 1 1 1 1 1 1 1 1

.. 4 10.0 0 0.0 0 0.0 0 0.0 0.5 0.0 1.5 0.0 2.5 0.0 3.5 0.0 4.5 0.0 5.5 0.0 6.5 0.0 7.5 0.0 8.5 0.0 9.5 0.0 1 2 3 4 5 6 7 8 9 10 12

B U L K

D A T A

.. 5 .. 6 1.1468-7 10 10 10.0 0.0 10 10 10.0 0.0 10 10 10.0 0.0 0.0 0.0

..

11

E C H O 7

..

8

..

10.0

0.0

0.0

0.0

20.0 0.5

0.0 0.0

1 10.0 1 10.0 1 10.0 1.0

9

..

10

+CA1 +CA2 +CA3 +CRD1

0.0

0.0

1.5

0.0

1.0

+CRD2

0.0

0.0

2.5

0.0

1.0

+CRD3

0.0

0.0

3.5

0.0

1.0

+CRD4

0.0

0.0

4.5

0.0

1.0

+CRD5

0.0

0.0

5.5

0.0

1.0

+CRD6

0.0

0.0

6.5

0.0

1.0

+CRD7

0.0

0.0

7.5

0.0

1.0

+CRD8

0.0

0.0

8.5

0.0

1.0

+CRD9

0.0

1.0

+CRD10

0.0

0.0

9.5

2 3 4 5 6 7 8 9 10 11 13

13 14 15 16 17 18 19 20 21 22 24

12 13 14 15 16 17 18 19 20 21 23

.

Flutter Analysis Sample Problems

76-

39404142434445464748495051525354555657585960616263646566676869707172737475CQUAD4 7778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129-

CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 49 1 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 CQUAD4 EIGR +ER

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 53

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 20 MAX

13 14 15 16 17 18 19 20 21 23 24 25 26 27 28 29 30 31 32 34 35 36 37 38 39 40 41 42 43 45 46 47 48 49 50 51 52 54

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 AGIV

14 15 16 17 18 19 20 21 22 24 25 26 27 28 29 30 31 32 33 35 36 37 38 39 40 41 42 43 44 46 47 48 49 50 51 52 53

25 26 27 28 29 30 31 32 33 35 36 37 38 39 40 41 42 43 44 46 47 48 49 50 51 52 53 54 55 57 58 59 60 61 62 63 64

24 25 26 27 28 29 30 31 32 34 35 36 37 38 39 40 41 42 43 45 46 47 48 49 50 51 52 53 54 56 57 58 59 60 61 62 63

66 68 69 70 71 72 73 74 75 76 77 79 80 81 82 83 84 85 86 87 88 90 91 92 93 94 95 96 97 98 99 101 102 103 104 105 106 107 108 109 110 112 113 114 115 116 117 118 119 120 121

65 67 68 69 70 71 72 73 74 75 76 78 79 80 81 82 83 84 85 86 87 89 90 91 92 93 94 95 96 97 98 100 101 102 103 104 105 106 107 108 109 111 112 113 114 115 116 117 118 119 120 15

+ER

NX Nastran Aeroelastic Analysis User’s Guide

7-127

65 54 56 57 58 59 60 61 62 63 64 65 67 68 69 70 71 72 73 74 75 76 78 79 80 81 82 83 84 85 86 87 89 90 91 92 93 94 95 96 97 98 100 101 102 103 104 105 106 107 108 109 0.

64 55 57 58 59 60 61 62 63 64 65 66 68 69 70 71 72 73 74 75 76 77 79 80 81 82 83 84 85 86 87 88 90 91 92 93 94 95 96 97 98 99 101 102 103 104 105 106 107 108 109 110 1000.

Chapter 7

Flutter Analysis Sample Problems

130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160-

7-128

GRID 161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220-

FLFACT FLFACT FLFACT +FL3 FLUTTER GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID

1 2 3 30000. 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

26

2.0

NX Nastran Aeroelastic Analysis User’s Guide

0.5 2.0 21600. 31200. PK

3.0 22200.

22800.

24000.

1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0

2 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0

3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

3.0

0.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0

26400.

28800. 4

29400.

+FL3

Flutter Analysis Sample Problems

246-

221222223224225226227228229230231232233234235236237238239240241242243244245GRID 247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303-

GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID 112 GRID GRID GRID GRID GRID GRID GRID GRID GRID MAT1 MKAERO1 +MK PAERO1 PARAM PARAM PARAM PARAM PARAM PSHELL SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SET1 SPC1 SPC1 SPC1 SPC1 +SP1 +SP2 SPC1 SPC1 SPLINE2 +SPL1 SPLINE2 +SPL2 SPLINE2 +SPL3 SPLINE2 +SPL4 SPLINE2 +SPL5 SPLINE2 +SPL6 SPLINE2 +SPL7 SPLINE2 +SPL8 SPLINE2 +SPL9 SPLINE2 +SPL10 ENDDATA TOTAL COUNT=

87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111

7.0 7.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0 10.0

9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0

10.0 1.0 0.0 113 10.0 2.0 114 10.0 3.0 115 10.0 4.0 116 10.0 5.0 117 10.0 6.0 118 10.0 7.0 119 10.0 8.0 120 10.0 9.0 121 10.0 10.0 10 1.03+7 3.9+6 2.0 3.0 0.001 0.1 0.2 0.4 1 COUPMASS1 GRDPNT 1 OPPHIPA 1 VREF 12.0 WTMASS .0025907 1 10 .041 10 1001 1 THRU 22 2001 12 THRU 33 3001 23 THRU 44 4001 34 THRU 55 5001 45 THRU 66 6001 56 THRU 77 7001 67 THRU 88 8001 78 THRU 99 9001 89 THRU 110 10001 100 THRU 121 1 1 1 1 2 1 110 1 3 1 THRU 1 3 12 22 45 55 56 66 89 99 100 110 1 3 111 THRU 1 6 1 THRU 1001 1001 1001 1091 -1.0 -1.0 2001 1001 1002 1092 -1.0 -1.0 3001 1001 1003 1093 -1.0 -1.0 4001 1001 1004 1094 -1.0 -1.0 5001 1001 1005 1095 -1.0 -1.0 6001 1001 1006 1096 -1.0 -1.0 7001 1001 1007 1097 -1.0 -1.0 8001 1001 1008 1098 -1.0 -1.0 9001 1001 1009 1099 -1.0 -1.0 10001 1001 1010 1100 -1.0 -1.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 .1 +MK

11 23 67

33 77

34 78

44 88

+SP1 +SP2

121 121 1001

0.0

1.0

1

+SPL1

2001

0.0

1.0

2

+SPL2

3001

0.0

1.0

3

+SPL3

4001

0.0

1.0

4

+SPL4

5001

0.0

1.0

5

+SPL5

6001

0.0

1.0

6

+SPL6

7001

0.0

1.0

7

+SPL7

8001

0.0

1.0

8

+SPL8

9001

0.0

1.0

9

+SPL9

10001

0.0

1.0

10

+SPL10

304

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Listing 7-34. Sorted Bulk Data for Flutter Analysis of Square Panel Using ZONA51 Aerodynamics 3D PANEL FLUTTER PINNED EDGES ZONA51 AERO, EXAMPLE HA145HB POINT = KFREQ 0.2002 0.2116 0.2205 0.2183 0.2138 0.2121 0.2121 0.2122 0.2129 POINT = KFREQ 0.2595 0.2424 0.2284 0.2222 0.2162 0.2141 0.2140 0.2142 0.2149 POINT = KFREQ 0.2841 0.2778 0.2721 0.2624 0.2492 0.2445 0.2450 0.2464 0.2552 POINT = KFREQ 0.4462 0.4331 0.4205 0.3971 0.3558 0.3189 0.3098 0.3003 0.2772 POINT = KFREQ 0.1455 0.1454 0.1456 0.1470 0.1539 0.1737 0.1764 0.1760 0.1743 POINT = KFREQ 0.2854 0.2769 0.2688 0.2535 0.2245 0.1892 0.1834 0.1810 0.1780 POINT = KFREQ 0.2897 0.2824 0.2756 0.2632 0.2429 0.2275 0.2244 0.2215 0.2164 POINT = KFREQ 0.4630 0.4502 0.4381 0.4158

1 MACH NUMBER = 2.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 4.9941931E+00 1.8000000E+03 -1.5295196E-01 4.7267179E+00 1.8500000E+03 -1.6892217E-01 4.5343862E+00 1.9000000E+03 -2.9962501E-01 4.5802159E+00 2.0000000E+03 -4.7017121E-01 4.6775627E+00 2.2000000E+03 -6.4868200E-01 4.7153649E+00 2.4000000E+03 -7.4875242E-01 4.7157297E+00 2.4500000E+03 -7.6651222E-01 4.7127347E+00 2.5000000E+03 -7.8205627E-01 4.6973310E+00 2.6000000E+03 -8.0754602E-01 2 MACH NUMBER = 2.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 3.8529315E+00 1.8000000E+03 -7.3716998E-02 4.1252098E+00 1.8500000E+03 -5.8460318E-02 4.3790030E+00 1.9000000E+03 6.9779575E-02 4.5012388E+00 2.0000000E+03 2.3505199E-01 4.6263223E+00 2.2000000E+03 4.0527347E-01 4.6713800E+00 2.4000000E+03 4.9942601E-01 4.6723228E+00 2.4500000E+03 5.1601660E-01 4.6695371E+00 2.5000000E+03 5.3050166E-01 4.6535463E+00 2.6000000E+03 5.5418670E-01 3 MACH NUMBER = 2.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 3.5204258E+00 1.8000000E+03 -1.1996645E-01 3.6000168E+00 1.8500000E+03 -1.2269000E-01 3.6753762E+00 1.9000000E+03 -1.2527983E-01 3.8116384E+00 2.0000000E+03 -1.3001584E-01 4.0133109E+00 2.2000000E+03 -1.3746606E-01 4.0898609E+00 2.4000000E+03 -1.4209375E-01 4.0820804E+00 2.4500000E+03 -1.4296828E-01 4.0586076E+00 2.5000000E+03 -1.4400871E-01 3.9187410E+00 2.6000000E+03 -1.5543495E-01 4 MACH NUMBER = 2.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 2.2409444E+00 1.8000000E+03 -6.8533927E-02 2.3091853E+00 1.8500000E+03 -7.0626095E-02 2.3780580E+00 1.9000000E+03 -7.2728880E-02 2.5180054E+00 2.0000000E+03 -7.6967016E-02 2.8103240E+00 2.2000000E+03 -8.5559450E-02 3.1355634E+00 2.4000000E+03 -9.4068855E-02 3.2280867E+00 2.4500000E+03 -9.6002229E-02 3.3298919E+00 2.5000000E+03 -9.7581819E-02 3.6080966E+00 2.6000000E+03 -9.0925038E-02 5 MACH NUMBER = 3.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 6.8719449E+00 1.8000000E+03 -1.3304967E-01 6.8773971E+00 1.8500000E+03 -1.3403367E-01 6.8678694E+00 1.9000000E+03 -1.3483095E-01 6.8046999E+00 2.0000000E+03 -1.3596269E-01 6.4970131E+00 2.2000000E+03 -1.3780704E-01 5.7577105E+00 2.4000000E+03 -1.8923114E-01 5.6676235E+00 2.4500000E+03 -2.8943378E-01 5.6815982E+00 2.5000000E+03 -3.6082676E-01 5.7366500E+00 2.6000000E+03 -4.6060652E-01 6 MACH NUMBER = 3.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 3.5038400E+00 1.8000000E+03 -5.6072671E-02 3.6108031E+00 1.8500000E+03 -5.7603832E-02 3.7197495E+00 1.9000000E+03 -5.9102427E-02 3.9454696E+00 2.0000000E+03 -6.1949197E-02 4.4543200E+00 2.2000000E+03 -6.5977067E-02 5.2855530E+00 2.4000000E+03 -1.8683814E-02 5.4533815E+00 2.4500000E+03 7.9316705E-02 5.5237727E+00 2.5000000E+03 1.4844792E-01 5.6180801E+00 2.6000000E+03 2.4399443E-01 7 MACH NUMBER = 3.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 3.4520025E+00 1.8000000E+03 -7.4259356E-02 3.5412250E+00 1.8500000E+03 -7.6250061E-02 3.6290245E+00 1.9000000E+03 -7.8217961E-02 3.7997696E+00 2.0000000E+03 -8.2076333E-02 4.1175284E+00 2.2000000E+03 -8.9420311E-02 4.3949976E+00 2.4000000E+03 -9.6153036E-02 4.4568248E+00 2.4500000E+03 -9.7727045E-02 4.5153017E+00 2.5000000E+03 -9.9254772E-02 4.6215367E+00 2.6000000E+03 -1.0216923E-01 8 MACH NUMBER = 3.0000 DENSITY RATIO 1./KFREQ VELOCITY DAMPING 2.1599417E+00 1.8000000E+03 -3.8562585E-02 2.2210848E+00 1.8500000E+03 -3.9748330E-02 2.2823322E+00 1.9000000E+03 -4.0938303E-02 2.4051633E+00 2.0000000E+03 -4.3329265E-02

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168

FLUTTER SUMMARY = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 1.3766975E+02 -6.6152077E+01 8.6500464E+02 1.4950076E+02 -7.9337761E+01 9.3934100E+02 1.6005396E+02 -1.5065875E+02 1.0056487E+03 1.6679207E+02 -2.4636630E+02 1.0479855E+03 1.7965297E+02 -3.6611383E+02 1.1287930E+03 1.9441389E+02 -4.5731497E+02 1.2215386E+03 1.9844882E+02 -4.7787851E+02 1.2468907E+03 2.0262749E+02 -4.9783594E+02 1.2731461E+03 2.1142363E+02 -5.3637769E+02 1.3284139E+03 = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 1.7844838E+02 -4.1326641E+01 1.1212242E+03 1.7129987E+02 -3.1460678E+01 1.0763088E+03 1.6573328E+02 3.6331882E+01 1.0413329E+03 1.6971854E+02 1.2532655E+02 1.0663730E+03 1.8164279E+02 2.3126837E+02 1.1412954E+03 1.9624445E+02 3.0790619E+02 1.2330403E+03 2.0029247E+02 3.2469693E+02 1.2584747E+03 2.0450198E+02 3.4082715E+02 1.2849240E+03 2.1341290E+02 3.7155801E+02 1.3409128E+03 = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 1.9530289E+02 -7.3606873E+01 1.2271243E+03 1.9629016E+02 -7.5658478E+01 1.2333275E+03 1.9746184E+02 -7.7716675E+01 1.2406893E+03 2.0042395E+02 -8.1864525E+01 1.2593009E+03 2.0938774E+02 -9.0426689E+01 1.3156221E+03 2.2414758E+02 -1.0005964E+02 1.4083608E+03 2.2925346E+02 -1.0296876E+02 1.4404420E+03 2.3528502E+02 -1.0644688E+02 1.4783395E+03 2.5343005E+02 -1.2375327E+02 1.5923480E+03 = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 3.0681232E+02 -6.6058441E+01 1.9277588E+03 3.0601611E+02 -6.7898376E+01 1.9227560E+03 3.0518454E+02 -6.9729942E+01 1.9175311E+03 3.0339240E+02 -7.3359985E+01 1.9062708E+03 2.9901825E+02 -8.0373993E+01 1.8787871E+03 2.9236609E+02 -8.6401794E+01 1.8369905E+03 2.8990268E+02 -8.7434624E+01 1.8215123E+03 2.8677496E+02 -8.7914406E+01 1.8018604E+03 2.7524948E+02 -7.8624863E+01 1.7294436E+03 = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 1.0005164E+02 -4.1820374E+01 6.2864301E+02 1.0274933E+02 -4.3265602E+01 6.4559308E+02 1.0567273E+02 -4.4761272E+01 6.6396136E+02 1.1226706E+02 -4.7953690E+01 7.0539478E+02 1.2934222E+02 -5.5996593E+01 8.1268115E+02 1.5921822E+02 -9.4653191E+01 1.0003976E+03 1.6511876E+02 -1.5013969E+02 1.0374718E+03 1.6807413E+02 -1.9052391E+02 1.0560409E+03 1.7311963E+02 -2.5051071E+02 1.0877428E+03 = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 1.9622739E+02 -3.4566925E+01 1.2329331E+03 1.9570380E+02 -3.5416084E+01 1.2296433E+03 1.9510628E+02 -3.6226505E+01 1.2258889E+03 1.9362553E+02 -3.7683239E+01 1.2165852E+03 1.8865689E+02 -3.9103493E+01 1.1853662E+03 1.7344115E+02 -1.0180464E+01 1.0897629E+03 1.7160564E+02 4.2760830E+01 1.0782301E+03 1.7287634E+02 8.0623108E+01 1.0862141E+03 1.7677335E+02 1.3550227E+02 1.1106997E+03 = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 1.9917404E+02 -4.6465843E+01 1.2514475E+03 1.9954900E+02 -4.7801292E+01 1.2538033E+03 1.9998390E+02 -4.9141842E+01 1.2565360E+03 2.0104999E+02 -5.1840824E+01 1.2632344E+03 2.0408798E+02 -5.7332844E+01 1.2823226E+03 2.0858542E+02 -6.3008167E+01 1.3105809E+03 2.0997708E+02 -6.4466866E+01 1.3193250E+03 2.1148746E+02 -6.5945610E+01 1.3288149E+03 2.1489102E+02 -6.8974457E+01 1.3502002E+03 = 5.0000E-01 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 3.1831848E+02 -3.8563629E+01 2.0000541E+03 3.1815436E+02 -3.9728912E+01 1.9990229E+03 3.1798462E+02 -4.0896469E+01 1.9979563E+03 3.1762656E+02 -4.3236248E+01 1.9957065E+03

Flutter Analysis Sample Problems

0.3770 0.3445 0.3372 0.3302 0.3168

2.6523955E+00 2.9023838E+00 2.9654558E+00 3.0288160E+00 3.1565666E+00

2.2000000E+03 2.4000000E+03 2.4500000E+03 2.5000000E+03 2.6000000E+03

-4.8146792E-02 -5.2976876E-02 -5.4179590E-02 -5.5378061E-02 -5.7755798E-02

3.1682230E+02 3.1585501E+02 3.1557745E+02 3.1528149E+02 3.1462247E+02

-4.7921787E+01 -5.2568306E+01 -5.3714508E+01 -5.4851200E+01 -5.7086735E+01

1.9906533E+03 1.9845756E+03 1.9828318E+03 1.9809722E+03 1.9768313E+03

Listing 7-35. Output from Flutter Analysis of Square Panel Using ZONA51 Aerodynamics

7.11

Estimation of Dynamic Stability Derivatives (Example HA145I)

The use of oscillatory aerodynamic theory for the estimation of dynamic stability derivatives has been demonstrated by Rodden and Giesing (1970), and by Rodden, Bellinger, and Giesing (1984). The procedure demonstrated may be followed in NX Nastran to obtain the generalized aerodynamic forces for rigid body motions via a DMAP ALTER in SOL 145. The present example uses the model from Example HA144D, after first repeating the derivation of Rodden (1987) with corrections. The theory of Rodden and Giesing (1970) is outlined for use with NX Nastran first for the case of symmetrical plunging and pitching. The definitions of the longitudinal stability derivatives are given in the equations for the lift coefficient

Equation 7-11. and moment coefficient

Equation 7-12. The intercept values Cz0 and Cm0 will be omitted from further discussion since they cannot be estimated from oscillatory theory. Complex oscillatory aerodynamic coefficients are related to the real coefficients in harmonic motion via

Equation 7-13. and

Equation 7-14. where Re( ) denotes the real part of ( ). Harmonic plunging with amplitude h0 [θ = 0, α = h/V = (h0 /V)Re(iωeiωt)] is then considered and Eq. 7-11 through Eq. 7-14 lead to

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Flutter Analysis Sample Problems

Equation 7-15.

Equation 7-16. If harmonic pitching with amplitude α0 [θ = α = α0 Re(eiωt is next considered, Eq. 7-11 through Eq. 7-14 lead to

Equation 7-17.

Equation 7-18. The relationship between the generalized aerodynamic force matrices and the longitudinal stability derivatives requires the following information: a reference wing area, S, the mean aerodynamic chord (M.A.C.), , a reference chord, cr , for the reduced frequency, k = ωcr /2V, the plunge mode amplitude, h0 , and the pitch mode amplitude, α0 . These parameters allow nondimensionalizing the elements of the generalized aerodynamic influence matrices. For one (m,k) pair, the generalized influence matrix Qhh due to rigid body plunge and pitch appears as

Equation 7-19. The relationship of Qhh to the complex coefficients indicated by the following equations:

Equation 7-20.

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, and

is

Flutter Analysis Sample Problems

Equation 7-21.

Equation 7-22.

Equation 7-23. Substitution of Eq. 7-20 through Eq. 7-23 into Equations 5 through 10 of Rodden and Giesing (1970) yields the final relationship between the Qhh matrix elements and the stability derivatives.

Equation 7-24.

Equation 7-25.

Equation 7-26.

Equation 7-27.

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Equation 7-28.

Equation 7-29.

Equation 7-30.

Equation 7-31. where b = cr /2 and lm( ) denotes the imaginary part of ( ). The calculation of lateral-directional derivatives is similar to that for the longitudinal derivatives illustrated above. The derivation follows along the lines given by Rodden and Giesing (1970) and the preceeding derivation. The definitions of the lateral-directional stability derivatives are given in the equations for the side force coefficient

Equation 7-32. the rolling moment coefficient

Equation 7-33. and the yawing moment coefficient

7-134

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Flutter Analysis Sample Problems

Equation 7-34. Eq. 7-32 through Eq. 7-34 include -terms but not -terms because the -terms are first order in reduced freqency (k) but -terms are second order in k. The higher-order solution of Rodden and Giesing (1970) demonstrated the procedure to obtain the longitudinal second-order -terms. Following this procedure in the lateral-directional case will lead to the -terms. Again assuming that there are harmonic motions, first sidesway

then roll , in order to compare the and, finally, yaw [φ = 0, = −β = 0 Re(eiωt)], lateral-directional stability derivatives in Eq. 7-32 through Eq. 7-34 to the elements of the NX Nastran generalized aerodynamic force matrix QHHL that appears as

Equation 7-35. where the i-subscript of Qij denotes side force by i = 1, rolling moment by i = 2, and yawing moment by i = 3; the j-subscript of Qij denotes sideslip by j = 1, roll rate by j = 2, and yaw rate by j = 3. In the following, the amplitude of the lateral displacement is y0 , the amplitude of the bank angle is φ0 , and the amplitude of the yaw angle is 0 , and the reference span of the vehicle is b. The expressions for the derivatives are:

Sideslip Derivatives

Equation 7-36.

NX Nastran Aeroelastic Analysis User’s Guide

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Flutter Analysis Sample Problems

Equation 7-37.

Equation 7-38.

Equation 7-39.

Equation 7-40.

Equation 7-41.

Equation 7-42.

Equation 7-43.

7-136

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Flutter Analysis Sample Problems

Equation 7-44.

Rolling Derivatives

Equation 7-45.

Equation 7-46.

Equation 7-47.

Equation 7-48.

Equation 7-49.

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Chapter 7

Flutter Analysis Sample Problems

Equation 7-50.

Yawing Derivatives

Equation 7-51.

Equation 7-52.

Equation 7-53. Note that the present calculation of dynamic stability derivatives from an oscillatory solution at only one reduced frequency does not permit obtaining the -derivatives. The present example uses the model from Example HA144D, which is contained in the input file HA144D_MODEL.DAT and is shown in Listing 7-15. Since Example HA144D calculated stability derivatives with respect to NACA coordinates at GRID 90, it is necessary to force the center of gravity here to GRID 90 by adding the large weight CONM1 90 to GRID 90 chosen arbitrarily as 8.0 × 109 lbs. Furthermore, it is essential to align the rigid body modes with the NACA axes, and this is accomplished by assigning large principal moments of inertia to CONM1 90 in the amount of 1.0 × 1012 lb-in2. With both the large weight and large moments of inertia added, the calculated rigid body modes will be motions relative to the NACA axes, and the generalized aerodynamic forces will lead to the desired stability derivatives. The Bulk Data Section begins with the request INCLUDE HA144D_MODEL.DAT. The next entry is CONM1 90. This is followed by the AERO entry, which is only required here to prescribe the antisymmetric motion. Next, the MKAERO1 entry specifies the Mach Number m = 0.9 and the two low reduced frequencies k = 0.001 and 0.01; two frequencies are selected to illustrate the frequency dependence of the dynamic derivatives. The vibration analysis for the rigid body modes is performed by EIGR 10, which specifies the modified Givens method MGIV and requests only the three rigid body modes normalized on their maximum components. The PARAM,OPPHIPA,1 along with the DISP request in Case Control provides the modal output so that the amplitudes of the rigid body modes are displayed. Case Control Commands

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Flutter Analysis Sample Problems

The Case Control Section begins with three title commands. It then requests ECHO = BOTH, SPC = 1 for the antisymmetric boundary condition, DISP = 25 for the SET 25 = 1 through 400 structural grid point displacements to be printed, and the vibration analysis METHOD = 10. The Executive Control Section specifies 5 minutes CPU time, and the Flutter Solution Sequence SOL 145. Then it provides the Alter to print the generalized aerodynamic force matrix QHHL for the flight conditions listed on the MKAERO entry and to exit the execution. Output The input data are shown in Listing 7-36 and Listing 7-37. The output data required for the stability derivative calculations follow in Listing 7-38. The first output shown is the eigenvalue analysis. The rigid body modes are of interest and their frequencies are exact rather than computed zeros because of the SUPORT entry in the input. The modal amplitudes are required in the stability derivative calculation, and they are found from the eigenvectors. It should be noted that the eigenvectors are in the basic coordinate system, so the signs of the roll and yaw angles must be reversed to obtain them in the reference (NACA) coordinate system for use in the stability derivative calculations. The first eigenvector is the sidesway mode, and its amplitude is y0 = 1.0. The second eigenvector is the roll mode, and its amplitude is φ0 = −0.06666667. Finally, the third eigenvector is the yaw mode, and 0 = −0.05590732. The final output shown is the matrix QHHL for the three rigid modes, as in Eq. 7-35, printed by columns for the two reduced frequencies: the first three columns correspond to the lower frequency, k = 0.001, and the last three columns are for k = 0.01. The generalized aerodynamic forces are summarized in Table 7-3. Table 7-3. Complex Generalized Aerodynamic Forces for Antisymmetric Rigid Body Modes of Example HA144D Force Q11 Q21 Q31 Q12 Q22 Q32 Q13 Q23 Q33

k = 0.001 1.2752-05 - i2.8634-02 -1.4758-05 + i3.4945-03 1.7980-05 - i2.3189-02 -2.1631-05 - i4.2480-03 5.6038-05 - i5.9515-02 -1.8804-05 - i3.1072-03 -8.0042+00 - i3.5915-02 9.7683-01 + i9.2522-03 -6.4820+00 - i3.2783-02

k = 0.01 1.2748-03 - i2.8634-01 -1.4776-03 + i3.4947-02 1.7977-03 - i2.3189-01 -2.1650-03 - i4.2464-02 5.5982-03 - i5.9523-01 -1.8821-03 - i3.1060-02 -8.0034+00 - i3.5914-01 9.7541-01 + i9.2577-02 -6.4807+00 - i3.2782-01

The reference geometry ccr = 10.0 ft, b = 40.0 ft, S = 200 sq.ft, and Eq. 7-36 through Eq. 7-53 lead to the desired stability derivatives. The results are summarized in Table 7-4. The frequency dependence of the derivatives is seen by comparing the columns for the two frequencies. The static solution in the last column is taken from Example HA144D and has no estimates for the dynamic derivatives, which can only be obtained by using the oscillatory aerodynamic theory. Table 7-4. Lateral-Directional Static and Dynamic Stability Derivativesfor FSW Airplane Equation Derivative Value for k = 0.001 Value for k = 0.01 Static Results Eq. 7-36 -0.71585 -0.71585 CYβ -0.71584 Eq. 7-37 -0.71584 -0.71577

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Chapter 7

Flutter Analysis Sample Problems

Table 7-4. Lateral-Directional Static and Dynamic Stability Derivativesfor FSW Airplane Equation Derivative Value for k = 0.001 Value for k = 0.01 Static Results

CIβ

Cnβ

CYp

CIp Cnp

CYr CIr Cnr N A S T R A N

-0.079700

-0.079675

Eq. 7-39

-0.032761

-0.032763

Eq. 7-40

-0.032761

-0.032713

Eq. 7-41

-0.034589

-0.034631

Eq. 7-42

0.25924

0.25923

Eq. 7-43

0.25923

0.25918

Eq. 7-44

0.050251

0.050242

-

Eq. 7-45

0.079650

0.079620

0.079650

Eq. 7-46

-0.10195

-0.10148

-

Eq. 7-47

-0.41846

-0.41852

-0.41847

Eq. 7-48

-0.098504

-0.098406

-

Eq. 7-49

-0.026053

-0.026042

-0.026052

Eq. 7-50

0.039415

0.039451

-

Eq. 7-51 Eq. 7-52 Eq. 7-53

0.72330 0.042985 -0.27751

0.72331 0.042989 -0.27751

0.72330 0.042986 -0.27751

C O N T R O L D E C K ID NXN, HA145H HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145H

E X E C U T I V E

NX Nastran Aeroelastic Analysis User’s Guide

-0.032761 0.25923

E C H O

$$$$$$$$ $$$$$$$$ $ $ $ MODEL DESCRIPTION 30 DEG FORWARD SWEPT WING WITH $ $ AILERON, CANARD AND AFT SWEPT $ $ VERTICAL FIN AND RUDDER. $ $ BAR MODEL WITH DUMBBELL MASSES. $ $ $ $ SOLUTION ANTISYMMETRIC STATIC STABILITY $ $ DERIVATIVE ANALYSIS USING DOUBLET$ $ LATTICE AERODYNAMICS AT MACH 0.9. $ $ $ $ OUTPUT RIGID BODY MODES AND OSCILLATORY $ $ AERODYNAMIC MATRIX QHHL. $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ CPU TIME IN MINUTES SOL 145 $ FLUTTER COMPILE SUBDMAP=FLUTTER,SOUIN=MSCSOU,NOREF,NOLIST $ V3 ALTER 50 $ V3 MATPRN QHHL// $ PRINTS QHHL MATRIX EXIT $ CEND EXAMPLE HA145H: 30 DEG FWD SWEPT WING WITH CANARD & FIN PAGE ANTISYMMETRIC STABILITY DERIVATIVES HALF-SPAN MODEL, RIGID SIDESWAY, ROLL, AND YAW MOTIONS C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145H: 30 DEG FWD SWEPT WING WITH CANARD & FIN 2 SUBTI = ANTISYMMETRIC STABILITY DERIVATIVES 3 LABEL = HALF-SPAN MODEL, RIGID SIDESWAY, ROLL, AND YAW MOTIONS 4 ECHO = BOTH 5 SPC = 1 $ ANTISYMMETRIC CONSTRAINTS 6 SET 25 = 1 THRU 400 7 DISP = 25 $ PRINT GRID POINT DISPLACEMENTS 8 METHOD = 10 9 BEGIN BULK

7-140

-

Eq. 7-38

2

Flutter Analysis Sample Problems

EXAMPLE HA145H: 30 DEG FWD SWEPT WING WITH CANARD & FIN ANTISYMMETRIC STABILITY DERIVATIVES HALF-SPAN MODEL, RIGID SIDESWAY, ROLL, AND YAW MOTIONS I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. $ INCLUDE HA144D_MODEL.DAT $ $ * GRID POINT 90 MASS * $ $ THE CONM1 ENTRY DEFINES A CONCENTRATED MASS IN ITS GENERAL $ FORM. IT DEFINES A 6*6 SYMMETRIC MASS MATRIX AT A GEOMETRIC $ GRID POINT. LISTED ARE THE ELEMENTS OF THE 3*3 MASS $ PARTITION, THE 3*3 STATIC UNBALANCE PARTITION, AND THE 3*3 $ INERTIA PARTITION. IN THIS CASE, A LARGE MASS, WITH LARGE $ PRINCIPAL MOMENTS OF INERTIA, IS CHOSEN AT THE REFERENCE $ POINT TO ALIGN THE COMPUTED MODES WITH THE REFERENCE NACA $ COORDINATE SYSTEM. $ $CONM1 EID G CID M11 M21 M22 M31 M32 CONM1 90 90 8.+9 8.+9 $+M1 M33 M41 M42 M43 M44 M51 M52 M53 +M1 8.+9 1.+12 $+M2 M54 M55 M61 M62 M63 M64 M65 M66 +M2 1.+12 1.+12 $ $ $ * * * AERODYNAMIC DATA * * * $ $ $ (LB-FT-SEC SYSTEM) $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ REFERENCE LENGTHS PLUS SYMMETRY KEYS. SYMXZ = -1 INDICATES $ ANTISYMMETRICAL MOTION. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 10.0 2.378-3 -1 $ $ $ * AERODYNAMIC CONDITIONS * $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 CARD AND ITS CONTINUATION CARD WILL BE USED $ TO GENERATE GENERALIZED AERO FORCE MATRICES. $ MKAERO1 0.9 +MK 0.001 0.01 $ $ * * VIBRATION SOLUTION PARAMETERS * * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM; IN THIS CASE THE $ MODIFIED GIVENS METHOD. THE THREE RIGID BODY MODES ARE $ DESIRED, NORMALIZED ON THE MAXIMUM DISPLACEMENTS. $ EIGR 10 MGIV 3 +IGR MAX $ $ THE PARAM,OPPHIPA,1 PROVIDES THE VIBRATION MODES FOR THE $ MODAL PRINT AND PLOT REQUESTS. $ PARAM OPPHIPA 1 ENDDATA INPUT BULK DATA CARD COUNT = 398

PAGE

10 $

3

.

$ $ $ $ $ $ $ $ $ $ $ $ +M1 +M1 +M2 +M2 $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ +MK $ $ $ $ $ $ $ $ +IGR $ $ $

Listing 7-36. Input Files for Estimation of Dynamic Stability Derivatives EXAMPLE HA145I: 30 DEG FWD SWEPT WING WITH CANARD & FIN PAGE 12 ANTISYMMETRIC STABILITY DERIVATIVES HALF-SPAN MODEL, RIGID SIDESWAY, ROLL, AND YAW MOTIONS S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 1AELIST 2000 1119 1123 1127 1131 2AELIST 3000 3103 3107 3111 3115 3AERO 1 10.0 2.378-3 -1 4AESURF 517 AILERON 110 2000 5AESURF 518 RUDDER 301 3000 6CAERO1 1000 1000 2 4 1 7+CAC 10. 0. 0. 10. 10. 5. 0. 10. 8CAERO1 1100 1000 8 4 1 9+CAW 25. 0. 0. 10. 13.45299+20. 0. 10. 10CAERO1 3100 1000 4 4 1 11+CA1FI 30.7735 0. 10. 10. 25. 0. 0. 10. 12CBAR 100 100 90 99 0. 0. 1. 13CBAR 101 100 97 98 0. 0. 1. 14CBAR 102 100 98 90 0. 0. 1. 15CBAR 103 100 99 100 0. 0. 1. 16CBAR 110 101 100 110 0. 0. 1. 17CBAR 120 101 110 120 0. 0. 1.

NX Nastran Aeroelastic Analysis User’s Guide

..

10

+CAC +CAW +CA1FI

7-141

.

Chapter 7

71-

PBAR

Flutter Analysis Sample Problems

1819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970101 7273747576777879808182838485868788899091929394-

CBAR 310 301 100 310 CONM1 90 90 8.+9 +M1 8.+9 +M2 1.+12 CONM2 97 97 0 1500.0 CONM2 98 98 0 1500.0 CONM2 99 99 0 1500.0 CONM2 100 100 0 1500.0 CONM2 111 111 0 600.0 CONM2 112 112 0 400.0 CONM2 121 121 0 600.0 CONM2 122 122 0 400.0 CONM2 311 311 0 30.0 CONM2 312 312 0 20.0 CORD2R 1 0 12.5 0. +CRD1 20. 0. 0. CORD2R 2 0 30. 0. +CRD2 38.66025+5.0 0. CORD2R 100 0 15.0 0.0 +CRD100 0.0 0.0 0.0 CORD2R 110 0 26.7265 10.0 +CRD2A 36.7265 15.7735 0. CORD2R 300 0 30.0 0. +CRD2FI 20.0 0. 5.7735 CORD2R 301 0 32.5 0. +CRD2R 22.5 0. 5.7735 EIGR 10 MGIV +IGR MAX GRID 90 15. 0. GRID 97 0. 0. GRID 98 10. 0. GRID 99 20. 0. GRID 100 30. 0. GRID 110 27.11325+5. GRID 111 24.61325+5. GRID 112 29.61325+5. GRID 120 21.33975+15. GRID 121 18.83975+15. GRID 122 23.83975+15. GRID 310 32.88675+0. GRID 311 30.38675+0. GRID 312 35.38675+0. MAT1 1 1.44+9 5.40+8 MKAERO1 0.9 +MK 0.001 0.01 OMIT1 4 110 120 310 PAERO1 1000 PARAM GRDPNT 90 PARAM OPPHIPA 1 PARAM WTMASS .031081 PBAR 100 1 2.0 .173611 +PB1F 1.0 1.0 1.0 -1.0 +PB2F 0.0 1 1.5 0.173611+2.0 0.462963 +PB1W 0.5 3.0 0.5 -3.0 +PB2W 0.0 PBAR 301 1 .75 .086806 +PB1FI 0.5 3.0 0.5 -3.0 +PB2FI 0. RBAR 111 110 111 123456 RBAR 112 110 112 123456 RBAR 121 120 121 123456 RBAR 122 120 122 123456 RBAR 311 310 311 123456 RBAR 312 310 312 123456 SET1 1000 98 99 SET1 1100 99 100 111 SET1 3100 99 100 311 SPC1 1 35 97 98 SPC1 1 135 90 SPLINE2 1501 1000 1000 1007 +SPC 1. -1. SPLINE2 1601 1100 1100 1131 +SPW -1. -1. SPLINE2 3100 3100 3100 3115 +SP2FI -1. -1. SUPORT 90 246 ENDDATA TOTAL COUNT= 95

0.

0. 8.+9

1. +M1 +M2

1.+12 1.+12

0.

12.5

0.

10.

+CRD1

0.

30.

0.

10.

+CRD2

0.0

15.0

0.0

-10.0

+CRD100

0.

26.7265 10.0

-10.0

+CRD2A

0.

30.0

10.0

0.

+CRD2FI

0.

32.5

-10.

0.0

+CRD2R

3

+IGR

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5. 5. 5. +MK

0.15 -1.0

0.5 1.0

-1.0

-1.0

+PB1F +PB2F

-0.5

3.0

+PB1W -0.5

-3.0

+PB2W

1.0 -0.5

.231482 3.0 -0.5

-3.0

+PB1FI +PB2FI

112 312 99

121 100

1000

0.

1.

1

+SPC

1100

0.

1.

2

+SPW

3100

0.

1.

300

+SP2FI

122

Listing 7-37. Sorted Bulk Data for Estimation of Dynamic Stability Derivatives

7-142

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Flutter Analysis Sample Problems

EXAMPLE HA145H: 30 DEG FWD SWEPT WING WITH CANARD & FIN ANTISYMMETRIC STABILITY DERIVATIVES HALF-SPAN MODEL, RIGID SIDESWAY, ROLL, AND YAW MOTIONS MODE EXTRACTION EIGENVALUE NO. ORDER 1 3 0.0 2 2 0.0 3 1 0.0 4 4 1.189524E+03 5 5 1.512659E+03 6 7 4.028532E+03 7 6 7.604801E+03 8 8 6.609194E+04 9 9 1.532933E+05 10 12 2.790091E+05 EIGENVALUE = 0.000000E+00 CYCLES = 0.000000E+00 R E A POINT ID. TYPE T1 90 G 0.0 97 G 0.0 98 G 0.0 99 G 1.169039E-31 100 G 3.507895E-31 110 G 2.243618E-14 111 G 2.243618E-14 112 G 2.243618E-14 120 G 6.677991E-14 121 G 6.677991E-14 122 G 6.677991E-14 310 G 3.505192E-31 311 G 3.505192E-31 312 G 3.505192E-31 EIGENVALUE = 0.000000E+00 CYCLES = 0.000000E+00 POINT ID. TYPE T1 90 G 0.0 97 G 0.0 98 G 0.0 99 G 6.381248E-33 100 G 1.913569E-32 110 G 1.232810E-15 111 G 1.232810E-15 112 G 1.232810E-15 120 G 3.698430E-15 121 G 3.698430E-15 122 G 3.698430E-15 310 G 1.874572E-32 311 G 1.874572E-32 312 G 1.874572E-32 EIGENVALUE = 0.000000E+00 CYCLES = 0.000000E+00

PAGE

21

R E A L RADIANS

E I G E N V A L U E S CYCLES GENERALIZED GENERALIZED MASS STIFFNESS 0.0 0.0 2.486483E+08 0.0 0.0 0.0 1.381378E+08 0.0 0.0 0.0 9.714776E+07 0.0 3.448947E+01 5.489170E+00 0.0 0.0 3.889292E+01 6.190001E+00 0.0 0.0 6.347072E+01 1.010168E+01 0.0 0.0 8.720551E+01 1.387919E+01 0.0 0.0 2.570835E+02 4.091611E+01 0.0 0.0 3.915269E+02 6.231343E+01 0.0 0.0 5.282131E+02 8.406773E+01 0.0 0.0

L

E I G E N V E C T O R

T2 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00 1.000000E+00

N O .

T3 R1 R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -5.659632E-17 0.0 -1.758605E-16 -9.020562E-16 -1.816386E-16 -9.020562E-16 -1.816386E-16 -9.020562E-16 -1.816386E-16 -2.706169E-15 -1.799392E-16 -2.706169E-15 -1.799392E-16 -2.706169E-15 -1.799392E-16 2.659543E-34 -1.878151E-16 7.254929E-36 -1.878151E-16 5.246537E-34 -1.878151E-16

R E A L E I G E N V E C T O R N O . T2 T3 R1 R2 2.083331E-09 0.0 6.666667E-02 2.083331E-09 0.0 6.666667E-02 2.083331E-09 0.0 6.666667E-02 2.083331E-09 0.0 6.666667E-02 2.083329E-09 0.0 6.666667E-02 2.083330E-09 3.333333E-01 6.666667E-02 2.083330E-09 3.333333E-01 6.666667E-02 2.083329E-09 3.333333E-01 6.666667E-02 2.083331E-09 1.000000E+00 6.666667E-02 2.083332E-09 1.000000E+00 6.666667E-02 2.083331E-09 1.000000E+00 6.666667E-02 -3.333333E-01 2.246368E-34 6.666667E-02 -3.333333E-01 -7.020377E-35 6.666667E-02 -3.333333E-01 5.194774E-34 6.666667E-02 R E A L

E I G E N V E C T O R

N O .

1 R3 0.0 0.0 0.0 0.0 0.0 7.760559E-33 7.760559E-33 7.760559E-33 1.746126E-32 1.746126E-32 1.746126E-32 -1.034798E-34 -1.034798E-34 -1.034798E-34

0.0 1.734723E-17 2.602085E-18 -3.167247E-15 -4.518557E-15 -4.460078E-15 -4.460078E-15 -4.460078E-15 -4.419003E-15 -4.419003E-15 -4.419003E-15 -4.517339E-15 -4.517339E-15 -4.517339E-15

2 R3 0.0 0.0 0.0 0.0 0.0 -2.867730E-18 -2.867730E-18 -2.867730E-18 -6.452392E-18 -6.452392E-18 -6.452392E-18 -1.179362E-34 -1.179362E-34 -1.179362E-34

0.0 3.231174E-26 8.077936E-28 -1.068990E-16 -2.464865E-16 -2.465620E-16 -2.465620E-16 -2.465620E-16 -2.465620E-16 -2.465620E-16 -2.465620E-16 -2.487544E-16 -2.487544E-16 -2.487544E-16

3

POINT ID. TYPE T1 T2 T3 R1 R2 R3 90 G 0.0 -1.280440E-07 0.0 2.430115E-10 0.0 5.590732E-02 97 G 0.0 -8.386099E-01 0.0 2.430115E-10 0.0 5.590732E-02 98 G 0.0 -2.795367E-01 0.0 2.430115E-10 0.0 5.590732E-02 99 G -1.446326E-18 2.795365E-01 0.0 2.430114E-10 0.0 5.590732E-02 100 G -4.338979E-18 8.386096E-01 0.0 2.430113E-10 0.0 5.590732E-02 110 G -2.795366E-01 6.772192E-01 1.215056E-09 2.430113E-10 -1.045336E-26 5.590732E-02 111 G -2.795366E-01 5.374509E-01 1.215056E-09 2.430113E-10 -1.045336E-26 5.590732E-02 112 G -2.795366E-01 8.169875E-01 1.215056E-09 2.430113E-10 -1.045336E-26 5.590732E-02 120 G -8.386098E-01 3.544383E-01 3.645169E-09 2.430113E-10 -2.352006E-26 5.590732E-02 121 G -8.386098E-01 2.146700E-01 3.645169E-09 2.430113E-10 -2.352006E-26 5.590732E-02 122 G -8.386098E-01 4.942066E-01 3.645169E-09 2.430113E-10 -2.352006E-26 5.590732E-02 310 G -4.338979E-18 1.000000E+00 -1.348490E-33 2.430113E-10 7.099896E-34 5.590732E-02 311 G -4.338979E-18 8.602317E-01 4.264840E-34 2.430113E-10 7.099896E-34 5.590732E-02 312 G -4.338979E-18 1.139768E+00 -3.123464E-33 2.430113E-10 7.099896E-34 5.590732E-02 MATRIX QHHL (GINO NAME 101 ) IS A CMP D.P. 6 COLUMN X 3 ROW RECTANG MATRIX. COLUMN 1 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 1.2752D-05,-2.8634D-02 -1.4758D-05, 3.4945D-03 1.7980D-05,-2.3189D-02 COLUMN 2 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -2.1631D-05,-4.2480D-03 5.6038D-05,-5.9515D-02 -1.8804D-05,-3.1072D-03 COLUMN 3 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -8.0042D+00,-3.5915D-02 9.7683D-01, 9.2522D-03 -6.4820D+00,-3.2783D-02 COLUMN 4 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 1.2748D-03,-2.8634D-01 -1.4776D-03, 3.4947D-02 1.7977D-03,-2.3189D-01 COLUMN 5 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -2.1650D-03,-4.2464D-02 5.5982D-03,-5.9523D-01 -1.8821D-03,-3.1060D-02 COLUMN 6 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -8.0034D+00,-3.5914D-01 9.7541D-01, 9.2577D-02 -6.4807D+00,-3.2782D-01 THE NUMBER OF NON-ZERO TERMS IN THE DENSEST COLUMN = 3

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 7

Flutter Analysis Sample Problems

THE DENSITY OF THIS MATRIX IS 100.00 PERCENT.

Listing 7-38. Output for Estimation of Dynamic Stability Derivatives

7.12

Servoelastic Stability Analysis of a Missile (Example HA110A)

This example of servoelastic stability analysis uses Solution 110, Modal Complex Eigenvalues, rather than Solution 145 since no aerodynamic forces are included. This example of a typical air-to-air missile has been considered by Anon. (1971), Rodden and Harder (1975), and Rodden, Harder, and Bellinger (1979) but with an incorrect transfer function for one of the servo elements (the actuator). Rodden, Harder, and Bellinger (1979) also represented the structural flexibility by the general structural element GENEL. Rodden (1987) in Example HA75J corrected the actuator transfer function and used a beam idealization of the structure, while retaining all the other features of Rodden, Harder, and Bellinger (1979). The fuselage/flipper configuration was idealized as a uniform beam with an all-movable aft section. This was an oversimplification of a typical missile since the aft fuselage is never hinged and a very light flipper is usually hinged near its midchord. The missile was assumed to weigh M + 1000 lb, to be L = 150 in long, and to have a fundamental frequency of f1 = 450 Hz. The present example introduces further significant modifications and provides a suitably representative idealization of an air-to-air missile, and should be regarded as a replacement for the earlier analysis. The missile is modeled as a right-half-model while the flipper is modeled as a light weight (5 lb) plate represented by four CQUAD4 elements, each 0.500 in thick, and with densities adjusted to give a total weight of 5 lbs and a center of gravity 1.0 in forward of the hinge line, which is placed midway between GRIDs 9 and 10. The one-sided structural model and the servo system are discussed below. The structural model is discussed first. The structural grid points are numbered 1 through 10 as shown in Figure 7-20. A rate gyro, located halfway between GRIDs 4 and 5, is assigned the number 45 with the origin of coordinates placed there. GRID 11 is the flipper hinge line station. GRIDs 1 through 11 and GRID 45 are constrained to be free only in plunge and pitch, i.e., only the symmetrical case is being considered. CBAR, PBAR, and MAT1 entries represent the aluminum beam structure (assumed weightless), and the weight of the structure is modeled to be concentrated at the grid points via CONM2 entries. The bending stiffnesses, Iy and Iz , are found from the vibration analysis result of Timoshenko, Young, and Weaver (1974, p. 424), ω1 (M/LEI)1/2 = (4.730/L)2 and the desired fundamental frequency, f1 = ω1 /2π = 45.0 Hz; the one-side stiffnesses become Iy = Iz = 135.54 in4. A very large cross-sectional area and polar moment of inertia are selected arbitrarily. The first eight CONM2 entries give the weight of each as 50.0 lb at GRIDs 1 through 8. The weight of the vertical flippers in the cruciform configuration have been included by adding the weight of one flipper divided between GRIDS 9 and 10 according to its center of gravity location, and the weight at GRID 9 becomes 52.833 lbs and at GRID 10 becomes 52.167 lbs. The flipper is modeled by four equal size CQUAD4 elements with corner GRIDs 21 through 29 as shown in Figure 7-20. The flipper is connected to the fuselage at GRID 24, and an RBAR is extended from GRID 11 on the fuselage centerline to GRID 12 at the side of the fuselage coincident with GRID 24. Multipoint constraints connect GRIDs 12 and 24 together except for the pitch degree of freedom. The flipper rotation δ is defined by Scalar Point SPOINT 49 and is the difference between the pitch rotations (R2) of GRIDs 24 and 12; the sign convention for the flipper angle δ is positive with the trailing edge down. The flipper and its hinge are idealized to be very stiff, and this condition is realized by providing lateral constraints (T2) on GRIDs 21 and 27 and by choosing a large modulus of elasticity (1.0E+10) on the MAT1 10 and MAT1 20 Bulk Data entries; the densities of 0.0188889 and 0.0144444 on MAT1 10 and MAT1 20 lead to the desired weight and center of gravity for the flipper. PARAM,GRDPNT,45 generates

7-144

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

the inertial data for the fuselage/flipper combination relative to GRID 45. PARAM,WTMASS converts the weights to mass units. Structural damping is included as an equivalent frequency-dependent viscous damping through the table TABDMP1. Assuming that g1 = 0.03 in the first mode, g2 = 0.05 in the second mode, and g3 = 0.03 in the third mode. The vibration analysis below gives the first three undamped frequencies as f1 = 45.2, f2 = 126.7 and f3 = 251.0 Hz. The damping-frequency pairs provide the data necessary for the tabular damping format. The structural model including the grid, stiffness, inertial, and damping data is contained in input file MISSILE_STRUCT.DAT shown in Listing 7-39. Next, the servo system is considered. The block diagram shown in Figure 7-21 is an oversimplification of an air-to-air missile control system, but is adequate for illustrative purposes. The signals in Figure 7-21 denoted by ei generally represent voltages or quantities equivalent to voltages output by the various servo elements. The transfer functions are as follows. For the command error:

Equation 7-54. for the position potentiometer,

Equation 7-55. for the servoposition error,

Equation 7-56. for the actuator,

Equation 7-57.

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Figure 7-20. Uniform Missile-Flipper Idealization

Figure 7-21. Missile Servo Block Diagram and for the rate gyro,

Equation 7-58. in which p denotes the Laplace transform operator. The flipper rotation, δ, is considered as part of the servo system and is assigned extra point numbers EPOINT 50 (it is also SPOINT 49 when considered as part of the flipper structure). The signals e1 , e2 , e3 , and e4 , respectively, are given the extra point numbers 51, 52, 53, and 54. The slope of the beam, h’ = ∂h/∂x at the rate gyro, is the rotation (R2) of GRID 45. The NX Nastran transfer function TF entry assumes a single output and multiple inputs of the form

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Equation 7-59. The transfer functions are rewritten in this format as follows: For EPOINT 50 (δ)

Equation 7-60. for EPOINT 51 (e1 )

Equation 7-61. for EPOINT 52 (e2 )

Equation 7-62. for EPOINT 53 (e3 )

Equation 7-63. and for EPOINT 54 (e4 )

Equation 7-64. The numerical values assumed for the servo time constants are Ka = 50.0 deg per deg/s, Ta = 0.002 s, Kp = 1.0 deg/deg, ωg = 60.0 Hz = 376.991 rad/s, ζg = 0.70, and Kg = 0.13 deg per deg/s. The solution to the modal servoelastic eigenvalue problem by the QR-Transform method NX Nastran (HESS) requires a nonsingular mass matrix. This requirement is met by differentiating the servo equations a sufficient number of times so that all servo elements effectively become second-order systems and the mass matrix will no longer be singular; the servo equations here require six differentiations, which will add six zero roots to the eigenvalue analysis, and result in a 20th order eigenvalue problem with 20 real or complex conjugate roots. The modified servo equations become

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Equation 7-65.

Equation 7-66.

Equation 7-67.

Equation 7-68.

Equation 7-69. One additional transfer function is necessary to make the servo representation and the structural representation of the flipper angles the same, i.e., SPOINT 49 and EPOINT 50 should be identical. This can be done by introducing a stiff spring between the two that would approximate the actuator stiffness, i.e., the oil column stiffness, the cylinder breathing (radial) stiffness, and the piston rod stiffness. If the structural and servo representations of the flipper angles are denoted by δ49 and δ50 , respectively, then the transfer function can be written as

Equation 7-70. where Kf is selected large enough that the new frequency added to the system is above the range of interest and no damping is considered. The value Kf = 1.0E+8 adds the frequency ff = 1631.7 Hz, which is sufficiently higher than the third fuselage bending frequency of f3 = 251.0 Hz. The above equation is written as equaling zero but this is not actually the case. In NX Nastran the TF terms are added to the existing system of equations and, in this case, the TF terms are added to the flipper hinge-moment equation and give rise to the new frequency because of the stiffness Kf and the moment of inertia of the flipper. The above six equations are placed on the TF entries, which are contained in input file MISSILE_SERVO.DAT along with the definitions of the SPOINT and EPOINTs and are shown in Listing 7-40.

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The use of the additional transfer function above to make SPOINT 49 and EPOINT 50 approximately equal is a realistic approach to the problem since an actuator has only a finite stiffness. However, if it is desired to make SPOINT 49 and EPOINT 50 exactly equal, i.e., to assume a rigid actuator. This can be done by the Lagrange Multiplier technique discussed by Blakely (Version 68, Appendix K). This requires the addition of another scalar point, SPOINT 48, which can be interpreted as the actuator hinge moment. The structural and servo models are called into the Bulk Data Section by means of the INCLUDE entry. The vibration analysis for the three rigid body modes of fuselage plunge and pitch and the open loop flipper rotation along with the three flexible modes is carried out by the modified Householder eigenvalue method as specified on the EIGR entry. All six modes are included in the modal formulation via PARAM,LMODES, and the complex eigenvalues are requested on the EIGC entry by the QR-Transform method (HESS) with the eigenvectors to be normalized on their maximum components. ENDDATA completes the Bulk Data Section.

Case Control Section The Case Control Section begins with the three title entries. ECHO = BOTH requests both annotated and sorted Bulk Data be printed. MPC = 50 specifies the constraints on the flipper (QUAD4) GRIDs and the flipper rotation in terms of the pitch rotations of GRIDs 24 and 12. SDAMP = 1 specifies the modal damping. TFL = 1 specifies the set number for the transfer functions, the TF entries. METHOD = 14 calls for the modified Householder real eigenvalue (vibration) method on EIGR; SVEC = ALL requests the vibration modes. CMETHOD = 40 calls for the complex Hessenberg (QR-Transform) eigenvalue method on the ElGC entry. SDISP = ALL requests the modal participation factors and the extra point displacements for each eigenvalue. DISP = ALL requests the grid point displacements for each complex eigenvalue. The BEGIN BULK command completes the Case Control Section. The Executive Control Section begins with the identification ID NXN, HA110A. TIME 5 limits the computing time to 5.0 CPU minutes. SOL 110 specifies the modal complex eigenvalue solution sequence. The CEND statement completes the Executive Control Section.

Output The input data file for this example is shown in Listing 7-41, followed by the sorted Bulk Data entries in Listing 7-42, and representative output results in Listing 7-43. The results are discussed below. The weight analysis is the first output. The half-missile weighs 510.0 lbs and has a center of gravity 16.157 inches aft of the rate gyro location at Station 61.157. The vibration frequencies for the open loop are three computed zeroes and three fuselage frequencies slightly coupled with the overbalanced flipper, f1 = 45.17 Hz, f2 = 126.74 Hz and f3 = 251.04 Hz. The rigid body modes and the coupled bending modes are shown next, and the flipper coupling is evident. The 22 roots from the closed-loop servoelastic solution consist of four zero roots from the two rigid body modes, four complex conjugate pairs from the three coupled vibration modes (f1 = 45.13 Hz, g1 = 0.0254; f2 = 126.12 Hz, g2 = 0.0507; f3 = 249.34 Hz, g3 = 0.0786). The frequency is introduced (with no damping) by the flipper servo/structure connection transfer function (ff = 1631.7), two real roots from the actuator/potentiometer loop (p = −440.29 corresponding to Ta = 0.002 and p = −56.324 corresponding to Ka = 50.0), a complex conjugate pair from the rate-gyro (fg = 42.85 Hz and gg = 1.981. Note that these are coupled values that correspond closely to the uncoupled input values of ωg = 60.0 and ζg = 0.70), and the six zero roots artificially added to permit the use of HESS by making the mass matrix nonsingular. Closing the servo loop is seen to cause only slight changes in the open loop root characteristics. The modal displacement output also shows the same lack of coupling. The modal displacements

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include the amplitudes of the vibration modes and the extra points. The grid point displacement output (note that it is not normalized) shows the servoelastic mode shapes and includes the scalar point SPOINT 49. Note that the amplitudes for EPOINT 50 and SPOINT 49 are equal as expected for the lower modes where there are only small modal convergence errors. $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ * * * STRUCTURAL DATA * * * $ $ (LB-IN-SEC SYSTEM) $ $ * * GRID GEOMETRY * * $ $ GRID 1 - 10 (T3,R2) FUSELAGE POINTS $ GRID 45 (T3,R2) SERVO SENSOR POINT $ $ * FUSELAGE GRID * $ $ THE GRID ENTRY DEFINES THE LOCATION OF A STRUCTURAL GRID $ POINT. LISTED ARE ITS COORDINATE SYSTEM ID, ITS LOCATION, $ THE ID OF THE COORDINATE SYSTEM IN WHICH ITS DISPLACEMENTS $ ARE DEFINED, ITS PERMANENT SINGLE-POINT CONSTRAINTS AND $ ITS ASSOCIATED SUPERELEMENT ID. $ $ ID CP X1 X2 X3 CD PS SEID GRID 1 -52.5 0. 0. 246 GRID 2 -37.5 0. 0. 246 GRID 3 -22.5 0. 0. 246 GRID 4 -7.5 0. 0. 246 GRID 5 7.5 0. 0. 246 GRID 6 22.5 0. 0. 246 GRID 7 37.5 0. 0. 246 GRID 8 52.5 0. 0. 246 GRID 9 67.5 0. 0. 246 GRID 10 82.5 0. 0. 246 GRID 45 0.0 0. 0. 1246 GRID 21 60.0 10. 0. 26 GRID 22 60.0 20. 0. 6 GRID 23 60.0 30. 0. 6 GRID 24 75.0 10. 0. 6 GRID 25 75.0 20. 0. 6 GRID 26 75.0 30. 0. 6 GRID 27 90.0 10. 0. 26 GRID 28 90.0 20. 0. 6 GRID 29 90.0 30. 0. 6 GRID 11 75.0 0. 0. 246 GRID 12 75.0 10. 0. $ $ SID G C A G C A MPC 50 24 1 1.0 12 1 -1.0 MPC 50 24 2 1.0 12 2 -1.0 MPC 50 24 3 1.0 12 3 -1.0 MPC 50 24 4 1.0 12 4 -1.0 $ MPC 50 49 1.0 24 5 -1.0 +MPC50 12 5 1.0 $ $ * * STRUCTURAL STIFFNESS PROPERTIES * * $ $ * FUSELAGE STRUCTURE * $ $ THE CBAR ENTRY DEFINES A SIMPLE BEAM ELEMENT. LISTED ARE $ ITS PROPERTY ENTRY ID, THE TWO GRID POINTS JOINED BY THE $ BEAM AND COMPONENTS OF A VECTOR FROM THE FIRST POINT. $ THIS VECTOR DEFINES THE DIRECTION OF THE STRUCTURAL DE$ FLECTION OF THE POINT AND ITS POSITIVE SENSE. $ $ EID PID GA GB X1,GO X2 X3 CBAR 1 1 1 2 0.0 0.0 1.0 CBAR 2 1 2 3 0.0 0.0 1.0 CBAR 3 1 3 4 0.0 0.0 1.0 CBAR 4 1 4 45 0.0 0.0 1.0 CBAR 45 1 45 5 0.0 0.0 1.0 CBAR 5 1 5 6 0.0 0.0 1.0 CBAR 6 1 6 7 0.0 0.0 1.0 CBAR 7 1 7 8 0.0 0.0 1.0 CBAR 8 1 8 9 0.0 0.0 1.0 CBAR 9 1 9 11 0.0 0.0 1.0 CBAR 10 1 11 10 0.0 0.0 1.0 $ $ THE PBAR ENTRY DEFINES GEOMETRIC PROPERTIES OF THE BEAM. $ $ LISTED ARE ITS ASSOCIATED MATERIAL ENTRY ID, ITS CROSS SEC$ TIONAL AREA, AREA MOMENTS OF INERTIA, TORSIONAL MOMENT $ OF INERTIA AND NON-STRUCTURAL MASS PER UNIT AREA. THE $ OPTIONAL CONTINUATION ENTRY CONTAINS STRESS RECOVERY $ COEFFICIENTS, I.E., Y,Z COORDINATES WHERE STRESSES ARE $ TO BE COMPUTED. K1 AND K2 ARE AREA FACTORS FOR SHEAR $ CALCULATIONS AND I12 IS THE AREA CROSS PRODUCT OF INERTIA. $

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$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

+MPC50 $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

Flutter Analysis Sample Problems

$

$ PID MID A I1 I2 J NSM PBAR 1 1 1.0+8 67.77 67.77 1.0+8 $ $ * * PHYSICAL PROPERTIES * * $ $ THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES. LISTED $ ARE ITS ID, ITS ELASTIC MODULUS, SHEAR MODULUS, POISSONS $ RATIO, MASS DENSITY, TEMPERATURE EXPANSION COEFFICIENT, $ REFERENCE TEMPERATURE AND A STRUCTURAL DAMPING COEFFICIENT. $ $ MID E G NU RHO A TREF GE MAT1 1 10.3+6 0.33 0.0 $ $ $ * * MASS AND INERTIA PROPERTIES * * $ $ * FUSELAGE MASSES * $ $ THE CONM2 ENTRY DEFINES A CONCENTRATED MASS. LISTED ARE $ ITS ID, GRID LOCATION, COORDINATE SYSTEM TO LOCATE THE $ CENTER OF GRAVITY, THE MASS VALUE AND THE LOCATION OF $ THE CENTER OF GRAVITY RELATIVE TO THE GRID LOCATION. $ $ EID G CID M X1 X2 X3 CONM2 1 1 50. CONM2 2 2 50. CONM2 3 3 50. CONM2 4 4 50. CONM2 5 5 50. CONM2 6 6 50. CONM2 7 7 50. CONM2 8 8 50. CONM2 9 9 52.833 CONM2 10 10 52.167 $ $ * * FUSELAGE DAMPING CHARACTERISTICS * * $ $ THE TABDMP1 ENTRY DEFINES MODAL DAMPING AS A TABULAR $ FUNCTION OF FREQUENCY. THE DAMPING LEVELS ARE LINEAR $ BETWEEN THE FREQUENCY AND DAMPING PAIRS AND EXTRAPOLATED $ OUTSIDE THE TABULATED FREQUENCY RANGE. THERE IS NO $ PARAMETER, KDAMP,N; THUS KDAMP DEFAULTS TO 1. THIS $ SPECIFIES THAT THE DAMPING CONSTANTS ARE TO BE A PART $ OF THE DAMPING MATRIX RATHER THAN THE STIFFNESS MATRIX $ AS IS REQUIRED IN THE PK-FLUTTER METHOD. $ $ ID TYPE TABDMP1 1 G $ F1 G1 F2 G2 ETC ENDT +DMP1 0.0 0.03 45.2 0.03 126.7 0.05 251.0 0.08 +DMP2 1000.0 0.08 ENDT $ $ $ * * FLIPPER STRUCTURE * * $ CQUAD4 1 1 21 22 25 24 CQUAD4 2 1 22 23 26 25 CQUAD4 3 2 24 25 28 27 CQUAD4 4 2 25 26 29 28 $ $ $ PROPERTY AND MATERIAL BULK DATA ENTRIES $ PSHELL 1 10 .500 10 PSHELL 2 20 .500 20 $ MAT1 10 1.0+10 0.30 .0188889 MAT1 20 1.0+10 0.30 .0144444 $ THE RBAR ENTRY DEFINES A RIGID BAR. LISTED ARE THE GRID $ $ POINTS AT EACH END AND THE DEPENDENT AND INDEPENDENT DOFS $ AT EACH END. THE NUMBER OF INDEPENDENT DOFS AT THE TWO $ ENDS MUST EQUAL SIX. BY DEFAULT THOSE NOT DECLARED INDE$ PENDENT ARE MADE DEPENDENT. $ $ EID GA GB CNA CNB CMA CMB RBAR 11 11 12 123456 $ PARAM GRDPNT 45 $ $ THE PARAM,WTMASS,GINV CAUSES ALL THE STRUCTURAL MASSES AND $ MASS DENSITIES TO BE MULTIPLIED BY GINV (I.E., BY ONE OVER $ THE ACCELERATION OF GRAVITY). $ PARAM WTMASS .0025907 $ PARAM COUPMASS1 $

$ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ +DMP1 +DMP2 $ $

$ $ $ $ $

$ $ $ $

Listing 7-39. MISSILE_STRUCT.DAT Input File

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* * SERVO SYSTEM EXTRA POINTS AND TRANSFER FUNCTIONS * * $ $ $ * SCALAR AND EXTRA POINTS * $ $ THE SPOINT ENTRY DEFINES SCALAR POINTS OF THE CONTROL $ SYSTEM. $ SPOINT 49 $ $ THE EPOINT ENTRY DEFINES EXTRA POINTS OF THE SERVO $ MODEL. $ $ ID ID ID ID ETC EPOINT 50 51 52 53 54 $ $ * TRANSFER FUNCTIONS * $ $ THE TF ENTRY DEFINES A SECOND ORDER TRANSFER FUNCTION $ FOR THE GRID OR EXTRA POINT GD IN THE DOF COMPONENT CD $ (CD IS ZERO FOR EXTRA POINTS). IT LISTS GD, CD AND THE $ COEFFICIENTS B OF THE POLYNOMIAL IN THE LAPLACE TRANSFORM $ VARIABLE P THAT RELATES THE DISPLACEMENTS UD TO OTHER $ DISPLACEMENTS UI, AND IT LISTS THE CORRESPONDING GS,CS AND $ COEFFICIENTS AI FOR THE OTHER DISPLACEMENTS. $ $ $ SCALAR/EXTRA POINT SPRING (DELETED WHEN LAGRANGE MULTIPLIER IS USED) $ $ SID GD CD B0 B1 B2 TF 1 49 1.0+8 $ G(1) C(1) A0(1) A1(1) A2(1) ETC +4950 50 -1.0+8 $ $ LAGRANGE MULTIPLIER FOR RIGID ACTUATOR $ $ SPOINT 48 IS ADDED FOR LAGRANGE MULTIPLIER WITH RIGID ACTUATOR $SPOINT 48 $ $ NAME "0" IFO TIN TOUT POLAR NCOL $DMIG STIFF 0 6 1 0 $ NAME GJ CJ G1 C1 A1 B1 $DMIG STIFF 48 49 -1.0 $ G2 C2 A2 B2 ETC $+STIFF 50 +1.0 $ $ ACTUATOR $ $ NOTE: THIS IS NOT THE TRANSFER FUNCTION USED FOR THE $ ACTUATOR IN THE EXAMPLE OF NASA CR 3094, BUT HAS BEEN $ CORRECTED EXTENSIVELY, EVEN BEYOND THE ONE DISCUSSED IN $ THE FOOTNOTE ON P. 82 OF THAT REPORT. $ TF 1 50 1.0 0.002 +5053 53 -50.0 $ $ COMMAND ERROR $ TF 1 51 1.0 +511 54 -1.0 $ $ POSITION POTENTIOMETER $ TF 1 52 1.0 +550 50 -1.0 $ $ SERVOPOSITION ERROR $ TF 1 53 1.0 +531 51 -1.0 +532 52 1.0 $ $ RATE GYRO $ TF 1 54 1.0 3.7136-37.0362-6 +5445 45 5 -0.13

Listing 7-40. MISSILE_SERVO.DAT Input File

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$ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $ $ $

$ +4950

+STIFF

$ $ $ $ +5053 $ $ $ +511 $ $ $ +550 $ $ $ +531 +532 $ $ $ +5445

Flutter Analysis Sample Problems

N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA110A $$$$$$$$ AEROELASTIC ANALYSIS EXAMPLE HA110A $$$$$$$$ $ $ $ MODEL DESCRIPTION TYPICAL AIR-TO-AIR MISSILE $ $ BAR MODEL WITH 10 FUSELAGE STATIONS $ $ AND AN ACTIVE CONTROLS SYSTEM. $ $ $ $ SOLUTION MODAL COMPLEX EIGENVALUE SOLUTION $ $ FOR SERVOELASTIC STABILITY ANALYSIS $ $ $ $ OUTPUT EIGENVALUES AND EIGENVECTORS FOR $ $ USE IN EVALUATING STABILITY $ $ $ $$$$$$$$ $$$$$$$$ TIME 5 $ TIME IN CPU MINUTES SOL 110 $ MODAL COMPLEX EIGENVALUES CEND EXAMPLE HA110A: FLEXIBLE MISSILE WITH CONTROL SYSTEM PAGE 3 SYMMETRIC MOTIONS, NO AERODYNAMIC FORCES HALF-SPAN MODEL, SERVOELASTIC ANALYSIS C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA110A: FLEXIBLE MISSILE WITH CONTROL SYSTEM 2 SUBTI = SYMMETRIC MOTIONS, NO AERODYNAMIC FORCES 3 LABEL = HALF-SPAN MODEL, SERVOELASTIC ANALYSIS 4 ECHO = BOTH 5 MPC = 50 6 SDAMP = 1 $ MODAL DAMPING 7 TFL = 1 $ SERVO TRANSFER FUNCTION 8 $ K2PP = STIFF $ DMIG LAGRANGE MULTIPLIER (FOR RIGID ACTUATOR) 9 METHOD = 14 $ MODIFIED HOUSEHOLDER REAL EIGENVALUE METHOD 10 SVEC = ALL $ VIBRATION MODES 11 CMETHOD = 40 $ HESSENBERG COMPLEX EIGENVALUE METHOD 12 SDISP = ALL $ MODAL PARTICIPATION AND EPOINT AMPLITUDES 13 DISP = ALL $ GRID DISPLACEMENTS 14 BEGIN BULK EXAMPLE HA110A: FLEXIBLE MISSILE WITH CONTROL SYSTEM PAGE 4 SYMMETRIC MOTIONS, NO AERODYNAMIC FORCES HALF-SPAN MODEL, SERVOELASTIC ANALYSIS I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DEFINITIONS WILL BE FOUND. $ $ $ INCLUDE MISSILE_STRUCT.DAT $ INCLUDE MISSILE_SERVO.DAT $ $ SID METHOD F1 F2 NE ND +EIG EIGR 14 MHOU 6 +HOU $+EIG NORM G C +HOU MAX $ $ $ * * * SOLUTION SPECIFICATIONS * * * $ $ $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ $ USED IN THE ANALYSIS. $ $ $ PARAM LMODES 6 $ $ $ * * SERVOELASTIC STABILITY ANALYSIS * * $ $ $ $ THE EIGC ENTRY DEFINES DATA NEEDED TO PERFORM A COMPLEX $ $ EIGENVALUE ANALYSIS. THIS ONE SPECIFIES THAT THE UPPER $ $ HESSENBERG METHOD BE USED TO EXTRACT COMPLEX EIGENVALUES $ $ AND THAT THE EIGENVECTORS BE NORMALIZED ON THEIR MAXIMUM $ $ COMPONENTS. THE CONVERGENCE CRITERION DEFAULTS TO: $ $ 1.0-4 INVERSE POWER METHOD, $ $ 1.0-11 DETERMINANT METHOD, $ $ 1.0-15 HESSENBERG METHOD. $ $ WHEN THE HESSENBERG METHOD IS SPECIFIED ALL ENTRIES ON THE $ $ CONTINUATION ENTRY EXCEPT THE NUMBER OF DESIRED ROOTS (NDL) $ $ ARE IGNORED. $ $ $ $ SID METHOD NORM G C E EIGC 40 HESS MAX +EC $ ALFA1 OMEGA1 ALFB1 OMEGB1 L1 NE1 ND1 +EC 22 $ ENDDATA INPUT BULK DATA CARD COUNT = 295

Listing 7-41. Input Files for Servoelastic Stability Analysis of a Missile NX Nastran Aeroelastic Analysis User’s Guide

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EXAMPLE HA110A: FLEXIBLE MISSILE WITH CONTROL SYSTEM PAGE 10 SYMMETRIC MOTIONS, NO AERODYNAMIC FORCES HALF-SPAN MODEL, SERVOELASTIC ANALYSIS S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 1CBAR 1 1 1 2 0.0 0.0 1.0 2CBAR 2 1 2 3 0.0 0.0 1.0 3CBAR 3 1 3 4 0.0 0.0 1.0 4CBAR 4 1 4 45 0.0 0.0 1.0 5CBAR 5 1 5 6 0.0 0.0 1.0 6CBAR 6 1 6 7 0.0 0.0 1.0 7CBAR 7 1 7 8 0.0 0.0 1.0 8CBAR 8 1 8 9 0.0 0.0 1.0 9CBAR 9 1 9 11 0.0 0.0 1.0 10CBAR 10 1 11 10 0.0 0.0 1.0 11CBAR 45 1 45 5 0.0 0.0 1.0 12CONM2 1 1 50. 13CONM2 2 2 50. 14CONM2 3 3 50. 15CONM2 4 4 50. 16CONM2 5 5 50. 17CONM2 6 6 50. 18CONM2 7 7 50. 19CONM2 8 8 50. 20CONM2 9 9 52.833 21CONM2 10 10 52.167 22CQUAD4 1 1 21 22 25 24 23CQUAD4 2 1 22 23 26 25 24CQUAD4 3 2 24 25 28 27 25CQUAD4 4 2 25 26 29 28 26EIGC 40 HESS MAX 27+EC 22 28EIGR 14 MHOU 6 29+HOU MAX 30EPOINT 50 51 52 53 54 31GRID 1 -52.5 0. 0. 246 32GRID 2 -37.5 0. 0. 246 33GRID 3 -22.5 0. 0. 246 34GRID 4 -7.5 0. 0. 246 35GRID 5 7.5 0. 0. 246 36GRID 6 22.5 0. 0. 246 37GRID 7 37.5 0. 0. 246 38GRID 8 52.5 0. 0. 246 39GRID 9 67.5 0. 0. 246 40GRID 10 82.5 0. 0. 246 41GRID 11 75.0 0. 0. 246 42GRID 12 75.0 10. 0. 43GRID 21 60.0 10. 0. 26 44GRID 22 60.0 20. 0. 6 45GRID 23 60.0 30. 0. 6 46GRID 24 75.0 10. 0. 6 47GRID 25 75.0 20. 0. 6 48GRID 26 75.0 30. 0. 6 49GRID 27 90.0 10. 0. 26 50GRID 28 90.0 20. 0. 6 51GRID 29 90.0 30. 0. 6 52GRID 45 0.0 0. 0. 1246 53MAT1 1 10.3+6 0.33 0.0 54MAT1 10 1.0+10 0.30 .0188889 55MAT1 20 1.0+10 0.30 .0144444 56MPC 50 24 1 1.0 12 1 -1.0 57MPC 50 24 2 1.0 12 2 -1.0 58MPC 50 24 3 1.0 12 3 -1.0 59MPC 50 24 4 1.0 12 4 -1.0 60MPC 50 49 1.0 24 5 -1.0 61+MPC50 12 5 1.0 62PARAM COUPMASS1 63PARAM GRDPNT 45 64PARAM LMODES 6 65PARAM WTMASS .0025907 66PBAR 1 1 1.0+8 67.77 67.77 1.0+8 67PSHELL 1 10 .500 10 68PSHELL 2 20 .500 20 69RBAR 11 11 12 123456 70SPOINT 49 71TABDMP1 1 G +DMP1 72+DMP1 0.0 0.03 45.2 0.03 126.7 0.05 251.0 73+DMP2 1000.0 0.08 ENDT 74TF 1 49 1.0+8 75+4950 50 -1.0+8 76TF 1 50 1.0 0.002 77+5053 53 -50.0 78TF 1 51 1.0 79+511 54 -1.0 80TF 1 52 1.0 81+550 50 -1.0 82TF 1 53 1.0 83+531 51 -1.0 84+532 52 1.0 85TF 1 54 1.0 3.7136-37.0362-6

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..

9

..

10

+EC +HOU

+MPC50

0.08

+DMP2 +4950 +5053 +511 +550 +531 +532 +5445

.

Flutter Analysis Sample Problems

86-

+5445 45 ENDDATA TOTAL COUNT=

5

-0.13

87

Listing 7-42. Sorted Bulk Data Entries for Servoelastic Stability Analysis of a Missile EXAMPLE HA110A: FLEXIBLE MISSILE WITH CONTROL SYSTEM PAGE 12 SYMMETRIC MOTIONS, NO AERODYNAMIC FORCES HALF-SPAN MODEL, SERVOELASTIC ANALYSIS O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 45 M O * 5.100000E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -9.999990E+01 * 0.000000E+00 5.100000E+02 0.000000E+00 0.000000E+00 0.000000E+00 8.240004E+03 * 0.000000E+00 0.000000E+00 5.100000E+02 9.999990E+01 -8.240004E+03 0.000000E+00 * 0.000000E+00 0.000000E+00 9.999990E+01 2.166665E+03 -7.399991E+03 0.000000E+00 * 0.000000E+00 0.000000E+00 -8.240004E+03 -7.399991E+03 1.096032E+06 0.000000E+00 * -9.999990E+01 8.240004E+03 0.000000E+00 0.000000E+00 0.000000E+00 1.098199E+06 S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 5.100000E+02 0.000000E+00 1.960782E-01 0.000000E+00 Y 5.100000E+02 1.615687E+01 0.000000E+00 0.000000E+00 Z 5.100000E+02 1.615687E+01 1.960782E-01 0.000000E+00 I(S) * 2.147057E+03 5.784306E+03 0.000000E+00 * * 5.784306E+03 9.628992E+05 0.000000E+00 * * 0.000000E+00 0.000000E+00 9.650463E+05 * I(Q) * 9.629340E+05 * * 2.112233E+03 * * 9.650463E+05 * Q * 6.020274E-03 9.999819E-01 0.000000E+00 * * -9.999819E-01 6.020274E-03 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 *

* * * * * *

R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED NO. ORDER MASS STIFFNESS 1 5 7.130438E-10 2.670288E-05 4.249895E-06 4.623947E-01 3.297077E-10 2 3 2.192974E-08 1.480869E-04 2.356877E-05 1.776271E-02 3.895316E-10 3 2 5.170114E-07 7.190350E-04 1.144380E-04 4.765784E-03 2.463964E-09 4 1 8.055728E+04 2.838261E+02 4.517233E+01 5.257575E-01 4.235359E+04 5 4 6.341577E+05 7.963402E+02 1.267415E+02 7.764813E-01 4.924116E+05 6 6 2.487909E+06 1.577311E+03 2.510368E+02 7.707402E-01 1.917531E+06 7 7 6.871412E+06 2.621338E+03 4.171989E+02 0.0 0.0 8 8 1.533512E+07 3.916008E+03 6.232521E+02 0.0 0.0 9 9 2.929255E+07 5.412259E+03 8.613878E+02 0.0 0.0 10 10 4.846846E+07 6.961929E+03 1.108025E+03 0.0 0.0 EIGENVALUE = 7.130438E-10 CYCLES = 4.249895E-06 R E A L E I G E N V E C T O R N O . 1 POINT ID.

TYPE 1 2 3 4 5 6 7 8 9 10 11 21 22 23 24 25 26 27 28 29 45 EIGENVALUE = CYCLES =

POINT ID.

TYPE 1 2 3 4 5 6 7 8 9 10

T1 T2 T3 R1 G -4.330477E-31 1.000000E+00 G -3.562243E-27 8.907031E-01 G -7.519573E-22 7.814063E-01 G -7.997911E-22 6.721095E-01 G -1.976760E-22 5.628126E-01 G -2.245655E-22 4.535158E-01 G -2.052530E-22 3.442190E-01 G -8.489863E-22 2.349221E-01 G -4.185310E-22 1.256253E-01 G -4.022325E-22 1.632845E-02 G -4.099719E-22 7.097687E-02 G -1.147505E-20 -5.419256E-02 G 3.904461E-20 -4.771242E-20 -5.419256E-02 G -3.163136E-20 -9.167137E-20 -5.419256E-02 G G 1.447040E-20 2.581622E-20 7.097687E-02 G -4.416677E-21 5.001378E-20 7.097687E-02 G -4.005826E-20 1.961463E-01 G -3.960658E-21 4.952496E-20 1.961463E-01 G -7.579341E-22 6.481551E-20 1.961463E-01 G 6.174611E-01 2.192974E-08 2.356877E-05 R E A L E I G E N V E C T O T1 G G G G G G G G G G

T2 5.954298E-33 1.486326E-27 3.569638E-23 7.275592E-23 -1.589857E-23 -9.195103E-23 3.437265E-22 1.259616E-22 1.652412E-21 1.577636E-21

T3

R1 -8.629664E-02 -5.502931E-02 -2.376198E-02 7.505348E-03 3.877268E-02 7.004001E-02 1.013073E-01 1.325747E-01 1.638420E-01 1.951093E-01

R2

-3.983088E-17 -2.912840E-17 -4.044074E-17 -4.236520E-17 -4.033232E-17 -3.469447E-17 -2.125036E-17 -2.862294E-17 R R2

N O .

R3 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 7.286456E-03 -8.344629E-03 -8.344629E-03 -8.344629E-03 -8.344629E-03 -8.344629E-03 -8.344629E-03 -8.344629E-03 -8.344629E-03 -8.344629E-03 7.286456E-03 2 R3 -2.084489E-03 -2.084489E-03 -2.084489E-03 -2.084489E-03 -2.084489E-03 -2.084489E-03 -2.084489E-03 -2.084489E-03 -2.084489E-03 -2.084489E-03

NX Nastran Aeroelastic Analysis User’s Guide

7-155

Flutter Analysis Sample Problems

Chapter 7

11 21 22 23 24 25 26 27 28 29 45 EIGENVALUE = CYCLES =

G 1.620639E-21 G -1.928446E-19 G 3.598886E-19 G -3.771999E-19 G G 8.262685E-20 G 1.117723E-21 G -3.074556E-19 G 2.522337E-19 G -1.090884E-19 G 5.170114E-07 1.144380E-04

TYPE T1 T2 1 G 1.551449E-33 2 G -4.209431E-27 3 G 9.278257E-23 4 G 1.790012E-21 5 G 3.195201E-21 6 G 2.798366E-21 7 G 1.477114E-20 8 G 1.617997E-20 9 G 1.941320E-20 10 G 1.924964E-20 11 G 1.933127E-20 21 G -1.197645E-19 22 G -6.048157E-20 23 G 2.638072E-20 24 G 25 G 7.404986E-20 26 G 2.777415E-20 27 G -3.309639E-20 28 G -6.165333E-20 29 G 1.188998E-19 45 G EIGENVALUE = 8.055728E+04 CYCLES = 4.517233E+01

-6.461024E-19 -1.029055E-18 2.062826E-20 2.334186E-20 4.350086E-19 6.765885E-19 R E A L

POINT ID.

POINT ID.

TYPE 1 2 3 4 5 6 7 8 9 10 11 21 22 23 24 25 26 27 28 29 45 EIGENVALUE = CYCLES =

POINT ID.

TYPE 1 2 3 4 5 6 7 8 9 10 11 21 22 23 24 25 26 27 28 29 45 EIGENVALUE = CYCLES =

POINT ID.

7-156

TYPE 1

T1

T2 -5.504608E-24 -7.194693E-24 -5.496885E-24 -5.726971E-24 -3.209346E-22 -9.569880E-22 -1.609433E-21 -1.367960E-21 7.585179E-21 7.106629E-21 7.386534E-21 -1.325798E-18 2.458886E-18 -2.585523E-18

G G G G G G G G G G G G G G G G 5.770939E-19 G 5.896732E-21 G -2.104592E-18 G 1.721998E-18 G -7.449252E-19 G 6.341577E+05 1.267415E+02

T1 T2 G 2.062405E-23 G 2.695627E-23 G 2.059511E-23 G 2.145717E-23 G 9.830693E-22 G 3.016256E-21 G 5.054763E-21 G 5.346837E-21 G -1.021020E-20 G -8.920767E-21 G -9.691196E-21 G 3.941774E-18 G -7.528736E-18 G 7.850181E-18 G G -1.715975E-18 G 8.061839E-21 G 6.438479E-18 G -5.303716E-18 G 2.387814E-18 G 2.487909E+06 2.510368E+02 T1 G

T2 8.231770E-23

-2.147016E-19 -3.018290E-19 -7.217523E-20 -8.465474E-20 7.489331E-20 4.609465E-20

-4.446147E-18 -7.068179E-18 1.530867E-19 1.675153E-19 2.979970E-18 4.652544E-18

1.340713E-17 2.131988E-17 -5.175882E-19 -5.512987E-19 -9.133412E-18 -1.425661E-17

T3

NX Nastran Aeroelastic Analysis User’s Guide

-1.510294E-16 -1.476683E-16 -1.162265E-16 -5.811324E-17 -7.979728E-17

R1 -1.757270E-02 -1.090694E-02 -4.241187E-03 2.424570E-03 9.090326E-03 1.575608E-02 2.242184E-02 2.908760E-02 3.575335E-02 4.241911E-02 3.908623E-02 1.000000E+00 1.000000E+00 1.000000E+00 3.908623E-02 3.908623E-02 -9.218276E-01 -9.218276E-01 -9.218276E-01 5.757448E-03

N O .

R1 1.000000E+00 4.148082E-01 -1.164419E-01 -5.268358E-01 -7.543173E-01 -7.618728E-01 -5.486975E-01 -1.505693E-01 3.706050E-01 9.466017E-01 6.554550E-01 5.243307E-01 5.245903E-01 5.248336E-01

-2.434034E-17 4.301742E-17 1.230027E-16 4.250073E-17 3.209238E-17 -1.214306E-17 2.341877E-17 8.326673E-17 N O .

R1 1.000000E+00 -1.472577E-01 -9.070869E-01 -9.910961E-01 -4.264308E-01 4.090631E-01 9.860994E-01 9.207531E-01 1.782314E-01 -9.550123E-01 -3.635074E-01 -2.906659E-01 -2.918037E-01 -2.928701E-01 -3.643051E-01 -3.656536E-01 -4.365167E-01 -4.376543E-01 -4.387207E-01 -7.720971E-01

3.041461E-05 1.958227E-05 2.823276E-05 3.341751E-05 2.733380E-05 3.041434E-05 1.958200E-05 2.823249E-05

R1

R2 -8.031675E-01

N O .

R3 3.957338E-02 3.789160E-02 3.214863E-02 2.184270E-02 8.055649E-03 -7.057895E-03 -2.094803E-02 -3.141067E-02 -3.726979E-02 -3.895797E-02 -3.854272E-02 -8.743007E-03 -8.732853E-03 -8.741221E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742815E-03 -8.752967E-03 -8.744600E-03 1.527356E-02 5

R2

-1.332869E-04 -8.584264E-05 -1.237439E-04 -1.464265E-04 -1.197983E-04 -1.332776E-04 -8.583329E-05 -1.237345E-04

E I G E N V E C T O R

R3 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 -4.443838E-04 6.406092E-02 6.406092E-02 6.406092E-02 6.406092E-02 6.406092E-02 6.406092E-02 6.406092E-02 6.406092E-02 6.406092E-02 -4.443838E-04 4

R2

6.556371E-01 6.559448E-01 7.866180E-01 7.868776E-01 7.871209E-01 -6.664273E-01

-2.084489E-03 -5.470162E-02 -5.470162E-02 -5.470162E-02 -5.470162E-02 -5.470162E-02 -5.470162E-02 -5.470162E-02 -5.470162E-02 -5.470162E-02 -2.084489E-03 3

R2

E I G E N V E C T O R

T3

R E A L

1.794757E-01 1.794757E-01 1.000000E+00 1.000000E+00 1.000000E+00 2.313901E-02

E I G E N V E C T O R

T3

R E A L

-1.458794E-16 -1.090741E-16 -1.466926E-16

E I G E N V E C T O R

T3

R E A L

1.794757E-01 -6.410487E-01 -6.410487E-01 -6.410487E-01

N O .

R3 8.089692E-02 6.765771E-02 2.988964E-02 -1.844859E-02 -5.222651E-02 -5.267720E-02 -1.957072E-02 2.862209E-02 6.665598E-02 7.996662E-02 7.666872E-02 4.862184E-03 4.817660E-03 4.854299E-03 4.861692E-03 4.861689E-03 4.861686E-03 4.861201E-03 4.905717E-03 4.869073E-03 -3.879775E-02 6

R3 -1.101790E-01

Flutter Analysis Sample Problems

2 3 4 5 6 7 8 9 10 11 21 22 23 24 25 26 27 28 29 45 C O M P L E X E I ROOT EXTRACTION NO. ORDER 1 14 2 10 3 15 4 16 5 18 6 19 7 22 8 8 9 1 10 2 11 3 12 4 13 5 14 6 15 7 16 11 17 9 18 12 19 13 20 17 21 20 22 21

G G G G G G G G G G G G G G G G G G G G G E N

1.075918E-22 8.220220E-23 8.564438E-23 2.505074E-21 8.000345E-21 3.050360E-20 2.324334E-20 1.518076E-20 1.782623E-20 1.615786E-20 1.086764E-17 -1.977089E-17 2.081993E-17 -4.856614E-18 1.578528E-20 1.684636E-17 -1.388363E-17 5.933441E-18

3.651567E-17 5.801794E-17 -1.013482E-18 -9.828061E-19 -2.418948E-17 -3.770502E-17

S U M M A R Y EIGENVALUE (REAL) (IMAG) -4.402914E+02 -2.400192E-06 -2.684033E-04 -2.634747E-04 -2.666100E+02 -2.692103E+02 -3.600679E+00 -2.835854E+02 -2.010827E+01 -7.924460E+02 -6.156796E+01 -1.566627E+03 -9.602314E-01 -1.025234E+04 -3.855580E-06 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.205092E-19 0.0 -8.010432E-07 0.0 -5.632386E+01 1.132465E-08 2.652530E-04 2.636524E-04 -2.666100E+02 2.692103E+02 -3.600678E+00 2.835854E+02 -2.010828E+01 7.924460E+02 -6.156797E+01 1.566627E+03 -9.602517E-01 1.025234E+04

6.409355E-01 1.000000E+00 1.144860E-01 -9.401705E-01 -9.540685E-01 8.801114E-02 9.889024E-01 6.519110E-01 -7.906066E-01 1.096615E-02 8.755981E-03 8.892408E-03 9.020342E-03 1.106171E-02 1.122335E-02 1.319591E-02 1.333229E-02 1.346018E-02 -4.807200E-01

1.597737E-05 1.030083E-05 1.484093E-05 1.754442E-05 1.436554E-05 1.597302E-05 1.029643E-05 1.483648E-05

-6.846268E-02 2.339617E-02 8.016788E-02 4.396639E-02 -4.232252E-02 -8.031262E-02 -2.502116E-02 6.761730E-02 1.104467E-01 9.973577E-02 -1.480850E-04 -1.427361E-04 -1.471153E-04 -1.479976E-04 -1.479961E-04 -1.479947E-04 -1.479102E-04 -1.532562E-04 -1.488740E-04 7.443208E-02

V A L U E

COMPLEX EIGENVALUE = -4.402914E+02, -2.400192E-06 C O M P L E X E I G E N V E C T O R

FREQUENCY (CYCLES) 0.0 4.193330E-05 4.284615E+01 4.513402E+01 1.261217E+02 2.493364E+02 1.631710E+03 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.196158E-05 4.284615E+01 4.513402E+01 1.261217E+02 2.493364E+02 1.631710E+03 N O . 1 (REAL/IMAGINARY)

DAMPING COEFFICIENT 0.0 2.037412E+00 1.980682E+00 2.539396E-02 5.074987E-02 7.859939E-02 1.873195E-04 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -2.012142E+00 1.980682E+00 2.539396E-02 5.074991E-02 7.859941E-02 1.873235E-04 (SOLUTION SET)

POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 M -2.497556E-03 -2.188552E-01 1.000000E+00 2.999264E-03 -1.634230E-03 -7.067685E-04 -1.487078E-14 -6.476408E-14 0.0 -2.426186E-08 -5.267752E-13 -6.864233E-11 50 E 7.648042E-02 -3.943876E-03 7.648042E-02 -8.042429E-02 -3.943876E-03 -7.218346E-10 -6.130140E-09 -7.218345E-10 -5.408305E-09 -6.130140E-09 COMPLEX EIGENVALUE = -5.632386E+01, 1.132465E-08 C O M P L E X E I G E N V E C T O R N O . 16 (SOLUTION SET) (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 M -2.497555E-03 -2.188551E-01 1.000000E+00 1.595758E-04 -3.412922E-05 -1.222823E-05 1.138451E-15 -2.176241E-18 0.0 -2.040788E-13 -2.080226E-13 1.041746E-13 50 E 7.607064E-02 3.178404E-05 7.607064E-02 -7.603886E-02 3.178404E-05 -1.566420E-15 -1.156583E-11 -1.566420E-15 -1.156427E-11 -1.156583E-11 COMPLEX EIGENVALUE = -2.666100E+02, 2.692103E+02 C O M P L E X E I G E N V E C T O R N O . 18 (SOLUTION SET) (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 R2 R3 1 M -2.497556E-03 -2.188552E-01 1.000000E+00 3.265452E-03 -3.072697E-04 -1.955497E-05 9.728689E-11 7.029131E-09 0.0 -1.818878E-03 1.510850E-03 5.594630E-04 50 E 7.618017E-02 -3.343896E-01 7.618017E-02 -4.105697E-01 -3.343896E-01 -3.246300E-04 -2.582723E-02 -3.246300E-04 -2.550260E-02 -2.582723E-02 COMPLEX EIGENVALUE = -3.600678E+00, 2.835854E+02 C O M P L E X E I G E N V E C T O R N O . 19 (SOLUTION SET) (REAL/IMAGINARY) POINT ID. 1 M 50

E

TYPE T1 T2 -2.497556E-03 -2.188552E-01 1.829028E-10 1.321660E-08 6.747200E-02 -2.899413E-01 2.424742E-02 3.216839E-01

T3 1.000000E+00 0.0 6.747200E-02 2.424742E-02

COMPLEX EIGENVALUE = -2.010828E+01, 7.924460E+02 C O M P L E X E I G E N V E C T O R POINT ID. TYPE T1 T2 1 M -2.497556E-03 -2.188552E-01 6.544550E-11 4.546524E-09 50 E 2.254093E-02 -4.713148E-01 -5.560987E-03 4.647857E-01

T3 1.000000E+00 0.0 2.254093E-02 -5.560987E-03

R1 -2.827280E-01 8.137608E-01 -3.574133E-01 2.974365E-01

R2 9.925086E-04 8.620156E-06 -2.899413E-01 3.216839E-01

N O . 20 (REAL/IMAGINARY) R1 4.803889E-03 2.334489E-05 -4.938557E-01 4.703467E-01

R3 3.198957E-04 3.639347E-06

(SOLUTION SET)

R2 7.363515E-01 7.706185E-02 -4.713148E-01 4.647857E-01

R3 3.236009E-03 4.558427E-05

NX Nastran Aeroelastic Analysis User’s Guide

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Flutter Analysis Sample Problems

Chapter 7

COMPLEX EIGENVALUE = -6.156797E+01, 1.566627E+03 C O M P L E X E I G E N V E C T O R

N O . 21 (REAL/IMAGINARY)

(SOLUTION SET)

POINT ID. TYPE T1 T2 T3 R1 R2 1 M -2.497548E-03 -2.188552E-01 1.000000E+00 4.329962E-03 -9.209930E-03 8.575137E-10 2.393256E-09 0.0 1.278126E-05 -6.375129E-05 50 E -3.079677E-03 1.553737E-01 -3.079677E-03 1.584534E-01 1.553737E-01 6.233489E-03 -6.851864E-01 6.233489E-03 -6.914200E-01 -6.851864E-01 COMPLEX EIGENVALUE = -9.602517E-01, 1.025234E+04 C O M P L E X E I G E N V E C T O R N O . 22 (SOLUTION SET) (REAL/IMAGINARY) POINT ID. 1 M

TYPE T1 T2 T3 R1 R2 -2.497555E-03 -2.188552E-01 1.000000E+00 4.190826E-03 -6.873792E-03 5.625004E-10 3.656385E-10 0.0 3.482586E-06 -2.685362E-05 50 E 2.087124E-08 -5.213630E-05 2.087124E-08 -5.215717E-05 -5.213630E-05 -1.723798E-07 7.290859E-04 -1.723798E-07 7.292583E-04 7.290859E-04 COMPLEX EIGENVALUE = -4.402914E+02, -2.400192E-06 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY)

POINT ID.

T1

R3 -9.805289E-03 -1.235619E-04

1

T2 T3 R1 R2 -1.083938E-25 0.0 7.489002E-04 0.0 1.278907E-31 0.0 -2.420726E-08 0.0 2 G -1.461998E-25 0.0 -5.629367E-05 0.0 1.671571E-31 0.0 -1.010795E-08 0.0 3 G 8.674005E-23 0.0 -5.659797E-04 0.0 1.277201E-31 0.0 2.756923E-09 0.0 4 G 1.775974E-21 0.0 -9.380016E-04 0.0 1.330640E-31 0.0 1.277467E-08 0.0 5 G 3.194835E-21 0.0 -1.701960E-03 0.0 7.614002E-30 0.0 1.836589E-08 0.0 6 G 2.805597E-21 0.0 -2.984474E-03 0.0 2.266757E-29 0.0 1.854971E-08 0.0 7 G 1.466178E-20 0.0 -3.928910E-03 0.0 3.695131E-29 0.0 1.330585E-08 0.0 8 G 1.612525E-20 0.0 -3.169032E-03 0.0 3.159097E-29 0.0 3.584715E-09 0.0 9 G 1.908131E-20 0.0 -5.855745E-05 0.0 -1.850673E-28 0.0 -9.036422E-09 0.0 10 G 1.892866E-20 0.0 4.636232E-03 0.0 -1.736391E-28 0.0 -2.291156E-08 0.0 11 G 1.900418E-20 0.0 2.181968E-03 0.0 -1.803152E-28 0.0 -1.590313E-08 0.0 12 G 1.900418E-20 0.0 2.181968E-03 0.0 -1.803152E-28 0.0 -1.590313E-08 0.0 21 G -9.562992E-20 0.0 1.142474E+00 2.977505E-07 3.141828E-26 0.0 -1.272164E-08 -7.389415E-13 22 G -1.056906E-19 -1.342332E-19 1.142476E+00 1.917387E-07 -5.829608E-26 1.053583E-25 -1.272795E-08 -4.757642E-13 23 G 7.371334E-20 -1.734321E-19 1.142479E+00 2.764144E-07 6.129635E-26 1.674935E-25 -1.273386E-08 -6.859329E-13 24 G 1.900418E-20 0.0 2.181968E-03 0.0 -1.803152E-28 0.0 -1.590313E-08 0.0 25 G 6.389806E-20 -7.473301E-20 2.183750E-03 3.271227E-07 -1.366710E-26 -3.644330E-27 -1.590755E-08 -8.118980E-13 26 G 2.753392E-20 -8.779015E-20 2.186763E-03 2.676062E-07 -1.441535E-28 -3.996483E-27 -1.591503E-08 -6.640917E-13 27 G 5.551257E-21 0.0 -1.138109E+00 2.977376E-07 4.990157E-26 0.0 -1.908556E-08 -7.389347E-13 28 G -9.320137E-20 2.052590E-20 -1.138107E+00 1.917257E-07 -4.082310E-26 -7.063441E-26 -1.909187E-08 -4.757574E-13 29 G 1.324462E-19 -3.824061E-20 -1.138104E+00 2.764013E-07 1.766473E-26 -1.102837E-25 -1.909778E-08 -6.859259E-13 45 G 0.0 0.0 -1.246038E-03 0.0 0.0 0.0 1.620216E-08 0.0 49 S 7.633720E-02 -7.161011E-10 COMPLEX EIGENVALUE = -5.632386E+01, 1.132465E-08 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY)

R3 5.797695E-05 -9.526034E-10 4.508500E-05 -9.146568E-10 2.465909E-05 -7.816072E-10 3.262019E-05 -5.354376E-10 7.205617E-05 -1.984354E-10 8.844947E-05 1.741706E-10 1.953588E-05 5.137614E-10 -1.296807E-04 7.637837E-10 -2.748838E-04 8.995580E-10 -3.319702E-04 9.375694E-10 -3.177651E-04 9.282317E-10 -3.177651E-04 9.282317E-10 7.601943E-02 2.121329E-10 7.601953E-02 2.118862E-10 7.601945E-02 2.120895E-10 7.601943E-02 2.121305E-10 7.601943E-02 2.121305E-10 7.601943E-02 2.121305E-10 7.601943E-02 2.121282E-10 7.601933E-02 2.123749E-10 7.601941E-02 2.121716E-10 5.022671E-05 -3.756537E-10

POINT ID.

R3 -1.479742E-06 -3.637402E-14 -1.806386E-06 -2.893097E-14 -2.557101E-06 -1.033299E-14 -3.246149E-06 7.739848E-15 -3.850735E-06 1.380879E-14 -5.191948E-06 7.997781E-15 -8.073825E-06 -1.203629E-17 -1.206431E-05 -2.142057E-15 -1.543015E-05 7.923335E-16 -1.667756E-05

7-158

TYPE 1

R3 7.821034E-01 -6.378468E-02

G

TYPE 1

G

T1

2

G

3

G

4

G

5

G

6

G

7

G

8

G

9

G

10

G

T2 -2.588883E-27 5.408521E-36 -7.909576E-27 7.069103E-36 8.684570E-23 4.544866E-36 1.776084E-21 4.716624E-36 3.199059E-21 1.217354E-34 2.818697E-21 4.010289E-34 1.469562E-20 2.454413E-33 1.615384E-20 1.587308E-33 1.905397E-20 2.156946E-33 1.890659E-20

T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

R1 -1.048524E-03 -4.946324E-13 -1.024694E-03 1.376270E-14 -9.922117E-04 3.175221E-13 -9.482896E-04 3.263781E-13 -8.952499E-04 1.453462E-13 -8.290851E-04 -2.848581E-14 -7.317878E-04 -8.359321E-14 -5.813243E-04 -5.752371E-14 -3.729839E-04 -4.465355E-14 -1.290345E-04

R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

16

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

2.261988E-33 0.0 1.897892E-20 0.0 2.191321E-33 0.0 12 G 1.897892E-20 0.0 2.191321E-33 0.0 21 G -7.800983E-20 0.0 5.827085E-31 0.0 22 G -1.384515E-19 -7.479317E-20 -9.952408E-31 1.922333E-30 23 G 1.080768E-19 -7.895102E-20 1.063509E-30 3.051343E-30 24 G 1.897892E-20 0.0 2.191321E-33 0.0 25 G 5.614046E-20 -7.669982E-20 -2.667306E-31 -2.912152E-32 26 G 2.754104E-20 -8.983057E-20 -1.241045E-33 -2.183031E-32 27 G 3.353032E-20 0.0 8.450722E-31 0.0 28 G -1.162205E-19 -1.935120E-20 -6.944578E-31 -1.228067E-30 29 G 1.425033E-19 -1.004520E-19 2.734176E-31 -1.911623E-30 45 G 0.0 0.0 0.0 0.0 49 S 7.606830E-02 -1.567060E-15 COMPLEX EIGENVALUE = -2.666100E+02, 2.692103E+02 C O M P L E X E I G E N V E C T O R NO.

2.829673E-15 -1.636776E-05 2.315163E-15 -1.636776E-05 2.315163E-15 7.605193E-02 7.480109E-16 7.605194E-02 7.557579E-16 7.605194E-02 7.493879E-16 7.605193E-02 7.481027E-16 7.605193E-02 7.481035E-16 7.605193E-02 7.481043E-16 7.605193E-02 7.481945E-16 7.605193E-02 7.404489E-16 7.605193E-02 7.468207E-16 -3.529785E-06 1.271603E-14

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T1 T2 T3 R1 R2 -2.592190E-26 0.0 1.790101E-03 0.0 8.722577E-26 0.0 -7.573710E-04 0.0 2 G -3.840655E-26 0.0 2.991778E-04 0.0 1.140068E-25 0.0 -6.183906E-04 0.0 3 G 8.682240E-23 0.0 -1.113430E-03 0.0 8.710357E-26 0.0 -5.992157E-04 0.0 4 G 1.776060E-21 0.0 -2.314708E-03 0.0 9.075053E-26 0.0 -4.750968E-04 0.0 5 G 3.197775E-21 0.0 -3.114705E-03 0.0 3.470508E-24 0.0 2.017478E-04 0.0 6 G 2.814842E-21 0.0 -3.300114E-03 0.0 1.077365E-23 0.0 1.470021E-03 0.0 7 G 1.468902E-20 0.0 -2.705969E-03 0.0 2.762999E-23 0.0 2.537102E-03 0.0 8 G 1.614796E-20 0.0 -1.307723E-03 0.0 2.357021E-23 0.0 2.218243E-03 0.0 9 G 1.908021E-20 0.0 7.246010E-04 0.0 -2.072951E-23 0.0 -4.008310E-05 0.0 10 G 1.893097E-20 0.0 3.077627E-03 0.0 -1.643091E-23 0.0 -3.606947E-03 0.0 11 G 1.900439E-20 0.0 1.881618E-03 0.0 -1.903741E-23 0.0 -1.735262E-03 0.0 12 G 1.900439E-20 0.0 1.881618E-03 0.0 -1.903741E-23 0.0 -1.735262E-03 0.0 21 G -8.328386E-20 0.0 1.142233E+00 1.399600E-07 1.444694E-20 0.0 -1.387952E-03 -2.477582E-07 22 G -1.286132E-19 -9.253189E-20 1.142235E+00 9.012039E-08 -2.690829E-20 4.877233E-20 -1.390067E-03 -1.595502E-07 23 G 9.774972E-20 -1.071522E-19 1.142236E+00 1.299253E-07 2.821118E-20 7.752619E-20 -1.392049E-03 -2.300075E-07 24 G 1.900439E-20 0.0 1.881618E-03 0.0 -1.903741E-23 0.0 -1.735262E-03 0.0 25 G 5.843712E-20 -7.607556E-20 1.882456E-03 1.537726E-07 -6.359340E-21 -1.627450E-21 -1.736744E-03 -2.721955E-07 26 G 2.755703E-20 -8.915251E-20 1.883872E-03 1.257867E-07 1.028608E-23 -1.687463E-21 -1.739251E-03 -2.226772E-07 27 G 2.511169E-20 0.0 -1.138470E+00 1.399564E-07 2.298049E-20 0.0 -2.082884E-03 -2.477462E-07 28 G -1.093218E-19 -7.423877E-21 -1.138469E+00 9.011672E-08 -1.891260E-20 -3.275254E-20 -2.084999E-03 -1.595380E-07 29 G 1.394940E-19 -8.183152E-20 -1.138468E+00 1.299216E-07 8.282097E-21 -5.109657E-20 -2.086981E-03 -2.299952E-07 45 G 0.0 0.0 -2.778333E-03 0.0 0.0 0.0 -2.233180E-04 0.0 49 S 7.618120E-02 -2.185735E-04 COMPLEX EIGENVALUE = -3.600678E+00, 2.835854E+02 C O M P L E X E I G E N V E C T O R NO. 19 (REAL/IMAGINARY)

R3 1.001414E-04 -1.139710E-05 9.790190E-05 -5.001887E-06 8.895720E-05 -2.263948E-07 6.904637E-05 -2.275130E-05 3.511228E-05 -6.896113E-05 -1.241442E-05 -9.042781E-05 -6.720172E-05 -3.639847E-05 -1.172564E-04 8.637743E-05 -1.498873E-04 2.063258E-04 -1.603274E-04 2.534682E-04 -1.577486E-04 2.417379E-04 -1.577486E-04 2.417379E-04 7.602345E-02 2.316526E-05 7.602350E-02 2.308252E-05 7.602346E-02 2.315064E-05 7.602345E-02 2.316439E-05 7.602345E-02 2.316439E-05 7.602345E-02 2.316438E-05 7.602345E-02 2.316353E-05 7.602341E-02 2.324626E-05 7.602344E-02 2.317813E-05 5.396002E-05 -4.475635E-05

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

POINT ID.

R3 -1.114984E-02 3.220356E-02 -1.067415E-02 3.083503E-02 -9.058549E-03 2.616164E-02 -6.174589E-03 1.777486E-02

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

11

POINT ID.

G

TYPE 1

G

TYPE 1

G

2

G

3

G

4

G

T1

T2 1.603110E-24 -4.478957E-24 2.090788E-24 -5.854135E-24 8.844915E-23 -4.472673E-24 1.777755E-21 -4.659887E-24

T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-7.686022E-14 -2.533413E-04 -5.692393E-14 -2.533413E-04 -5.692393E-14 1.140526E+00 -4.568786E-14 1.140526E+00 -4.548995E-14 1.140526E+00 -4.530442E-14 -2.533413E-04 -5.692393E-14 -2.532862E-04 -5.678520E-14 -2.531930E-04 -5.655064E-14 -1.141032E+00 -6.813094E-14 -1.141032E+00 -6.793305E-14 -1.141032E+00 -6.774755E-14 -9.229031E-04 2.472414E-13

0.0 0.0 0.0 0.0 0.0 9.207038E-09 2.318413E-17 5.928639E-09 1.493396E-17 8.547071E-09 2.152586E-17 0.0 0.0 1.011551E-08 2.546790E-17 8.274772E-09 2.083903E-17 9.206732E-09 2.318181E-17 5.928331E-09 1.493161E-17 8.546760E-09 2.152349E-17 0.0 0.0

18 (REAL/IMAGINARY)

R1 -2.831762E-01 8.137664E-01 -1.183071E-01 3.375557E-01 3.134864E-02 -9.476005E-02 1.471075E-01 -4.287264E-01

R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

7-159

Flutter Analysis Sample Problems

Chapter 7

3.291689E-21 0.0 -2.611464E-22 0.0 3.095171E-21 0.0 -7.787041E-22 0.0 7 G 1.516623E-20 0.0 -1.309539E-21 0.0 8 G 1.655402E-20 0.0 -1.113062E-21 0.0 9 G 1.690278E-20 0.0 6.172488E-21 0.0 10 G 1.689297E-20 0.0 5.783084E-21 0.0 11 G 1.688478E-20 0.0 6.010846E-21 0.0 12 G 1.688478E-20 0.0 6.010846E-21 0.0 21 G 3.046981E-19 0.0 -1.078809E-18 0.0 22 G -8.483353E-19 1.208859E-18 2.000808E-18 -3.617852E-18 23 G 8.544631E-19 1.961706E-18 -2.103853E-18 -5.751412E-18 24 G 1.688478E-20 0.0 6.010846E-21 0.0 25 G -1.104869E-19 -1.208741E-19 4.695839E-19 1.245678E-19 26 G 2.588644E-20 -1.381110E-19 4.798656E-21 1.363091E-19 27 G 6.410983E-19 0.0 -1.712517E-18 0.0 28 G -6.134086E-19 -8.797584E-19 1.401198E-18 2.424816E-18 29 G 3.576554E-19 -1.443758E-18 -6.061487E-19 3.785798E-18 45 G 0.0 0.0 0.0 0.0 49 S 6.753141E-02 2.424893E-02 COMPLEX EIGENVALUE = -2.010828E+01, 7.924460E+02 C O M P L E X E I G E N

2.117417E-01 -6.138409E-01 2.147984E-01 -6.199821E-01 1.555295E-01 -4.464996E-01 4.328658E-02 -1.225159E-01 -1.048131E-01 3.015877E-01 -2.691540E-01 7.702962E-01 -1.860430E-01 5.333804E-01 -1.860430E-01 5.333804E-01 9.919035E-01 4.266772E-01 9.918290E-01 4.268885E-01 9.917592E-01 4.270864E-01 -1.860430E-01 5.333804E-01 -1.860952E-01 5.335286E-01 -1.861835E-01 5.337790E-01 -1.364001E+00 6.401151E-01 -1.364075E+00 6.403264E-01 -1.364145E+00 6.405243E-01 1.866488E-01 -5.423207E-01

POINT ID.

R1 7.373726E-01 7.704858E-02 -1.054548E-01 -1.130905E-02 -6.662506E-01 -6.985893E-02 -7.328522E-01 -7.638279E-02 -3.214700E-01 -3.292201E-02 2.937617E-01 3.146189E-02 7.231553E-01 7.598186E-02 6.799609E-01 7.099650E-02 1.347128E-01 1.377321E-02 -7.015582E-01 -7.360896E-02 -2.648552E-01 -2.799675E-02 -2.648552E-01 -2.799675E-02 9.289470E-01 -2.238662E-02 9.281110E-01 -2.247428E-02 9.273272E-01 -2.255646E-02 -2.648552E-01 -2.799675E-02 -2.654414E-01 -2.805821E-02 -2.664324E-01 -2.816212E-02 -1.458781E+00 -3.361982E-02 -1.459617E+00 -3.370747E-02 -1.460400E+00 -3.378964E-02 -5.741407E-01 -5.953671E-02

7-160

5

G

6

G

TYPE 1

G

T1

2

G

3

G

4

G

5

G

6

G

7

G

8

G

9

G

10

G

11

G

12

G

21

G

22

G

23

G

24

G

25

G

26

G

27

G

28

G

29

G

45

G

49

S

T2 1.542649E-23 1.592951E-24 2.015837E-23 2.082037E-24 1.022531E-22 1.590717E-24 1.792136E-21 1.657300E-24 3.929624E-21 7.586384E-23 5.061367E-21 2.327806E-22 1.850948E-20 3.908823E-22 2.016032E-20 4.130648E-22 1.161987E-20 -7.859475E-22 1.242837E-20 -6.864724E-22 1.192925E-20 -7.459126E-22 1.192925E-20 -7.459126E-22 2.853799E-18 3.042249E-19 -5.735305E-18 -5.810222E-19 5.944458E-18 6.058382E-19 1.192925E-20 -7.459126E-22 -1.220574E-18 -1.324441E-19 3.355632E-20 6.221175E-22 4.819680E-18 4.968799E-19 -4.058901E-18 -4.093069E-19 1.916669E-18 1.842625E-19 0.0 0.0 2.300459E-02 -5.537441E-03

T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 9.895986E-18 1.034739E-18 1.577633E-17 1.645429E-18 0.0 0.0 -4.604254E-19 -3.992893E-20 -4.982135E-19 -4.252499E-20 0.0 0.0 -6.809798E-18 -7.048708E-19 -1.069968E-17 -1.100251E-18 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -8.726239E-06 2.474912E-05 -5.618362E-06 1.593458E-05 -8.100262E-06 2.297370E-05 0.0 0.0 -9.587782E-06 2.719265E-05 -7.842335E-06 2.224219E-05 -8.726155E-06 2.474890E-05 -5.618278E-06 1.593436E-05 -8.100177E-06 2.297348E-05 0.0 0.0

V E C T O R NO. (REAL/IMAGINARY)

-2.321709E-03 6.555080E-03 1.923262E-03 -5.744046E-03 5.871098E-03 -1.704715E-02 8.894699E-03 -2.556061E-02 1.061862E-02 -3.032787E-02 1.112283E-02 -3.170137E-02 1.099872E-02 -3.136353E-02 1.099872E-02 -3.136353E-02 7.853016E-02 -7.114675E-03 7.852725E-02 -7.106413E-03 7.852966E-02 -7.113222E-03 7.853014E-02 -7.114597E-03 7.853014E-02 -7.114597E-03 7.853014E-02 -7.114597E-03 7.853011E-02 -7.114519E-03 7.853302E-02 -7.122780E-03 7.853062E-02 -7.115972E-03 -4.339341E-03 1.242896E-02

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R3 5.939576E-02 6.229968E-03 4.977396E-02 5.211592E-03 2.223305E-02 2.305168E-03 -1.322667E-02 -1.417518E-03 -3.828248E-02 -4.022480E-03 -3.896618E-02 -4.061497E-03 -1.477783E-02 -1.512306E-03 2.083767E-02 2.203797E-03 4.911562E-02 5.138846E-03 5.904742E-02 6.166501E-03 5.658634E-02 5.911881E-03 5.658634E-02 5.911881E-03 7.959129E-02 3.744778E-04 7.955857E-02 3.710472E-04 7.958550E-02 3.738703E-04 7.959093E-02 3.744399E-04 7.959093E-02 3.744397E-04 7.959092E-02 3.744394E-04 7.959057E-02 3.744020E-04 7.962328E-02 3.778321E-04 7.959636E-02 3.750086E-04 -2.826093E-02 -2.986077E-03

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

20

R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -9.794817E-05 -1.026989E-05 -6.308295E-05 -6.614266E-06 -9.093537E-05 -9.534599E-06 0.0 0.0 -1.076041E-04 -1.128232E-05 -8.803589E-05 -9.230589E-06 -9.794137E-05 -1.026918E-05 -6.307608E-05 -6.613546E-06 -9.092843E-05 -9.533872E-06 0.0 0.0

Flutter Analysis Sample Problems

COMPLEX EIGENVALUE = -6.156797E+01, 1.566627E+03 C O M P L E X E I G E N V E C T O R (REAL/IMAGINARY) POINT ID.

G

TYPE 1

G

2

G

3

G

4

G

5

G

6

G

7

G

8

G

9

G

10

G

11

G

12

G

21

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22

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23

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24

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25

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27

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21

T2 T3 R1 R2 R3 6.416716E-23 0.0 -6.342238E-01 0.0 -8.675142E-02 -5.251993E-24 0.0 5.117881E-02 0.0 7.023078E-03 2 G 8.386395E-23 0.0 5.033420E-01 0.0 -5.401032E-02 -6.864518E-24 0.0 -4.086718E-02 0.0 4.363041E-03 3 G 1.509254E-22 0.0 7.889610E-01 0.0 1.815576E-02 -5.244625E-24 0.0 -6.372834E-02 0.0 -1.493812E-03 4 G 1.842847E-21 0.0 9.549002E-02 0.0 6.295767E-02 -5.464241E-24 0.0 -7.246005E-03 0.0 -5.112027E-03 5 G 5.147958E-21 0.0 -7.354501E-01 0.0 3.489576E-02 -1.598521E-22 0.0 5.998602E-02 0.0 -2.800949E-03 6 G 9.044224E-21 0.0 -7.539517E-01 0.0 -3.265237E-02 -5.105040E-22 0.0 6.081914E-02 0.0 2.702796E-03 7 G 3.849986E-20 0.0 5.676656E-02 0.0 -6.272961E-02 -1.946005E-21 0.0 -5.683640E-03 0.0 5.123695E-03 8 G 3.427805E-20 0.0 7.637780E-01 0.0 -1.997513E-02 -1.482927E-21 0.0 -6.313745E-02 0.0 1.593741E-03 9 G 3.105240E-20 0.0 5.094069E-01 0.0 5.210206E-02 -9.675518E-22 0.0 -4.158856E-02 0.0 -4.317673E-03 10 G 3.296025E-20 0.0 -6.057641E-01 0.0 8.546915E-02 -1.136381E-21 0.0 5.050157E-02 0.0 -7.050401E-03 11 G 3.173597E-20 0.0 1.439241E-02 0.0 7.712429E-02 -1.029912E-21 0.0 -6.679200E-04 0.0 -6.366994E-03 12 G 3.173597E-20 0.0 1.439241E-02 0.0 7.712429E-02 -1.029912E-21 0.0 -6.679200E-04 0.0 -6.366994E-03 21 G 8.380040E-18 0.0 1.152228E+00 1.385521E-05 7.585504E-02 -6.934570E-19 0.0 -5.332671E-04 -1.010226E-06 9.023699E-06 22 G -1.552224E-17 2.834312E-17 1.152346E+00 8.931711E-06 7.585967E-02 1.261591E-18 -2.330052E-18 -5.418932E-04 -6.513125E-07 8.685488E-06 23 G 1.630886E-17 4.507268E-17 1.152457E+00 1.286906E-05 7.585588E-02 -1.328526E-18 -3.702105E-18 -5.499823E-04 -9.383741E-07 8.962375E-06 24 G 3.173597E-20 0.0 1.439241E-02 0.0 7.585511E-02 -1.029912E-21 0.0 -6.679200E-04 0.0 9.018157E-06 25 G -3.724140E-18 -8.639725E-19 1.447529E-02 1.521482E-05 7.585511E-02 3.098943E-19 6.467960E-20 -6.739621E-04 -1.109303E-06 9.018064E-06 26 G 3.983757E-20 -8.527413E-19 1.461546E-02 1.245703E-05 7.585511E-02 -1.007298E-21 6.272525E-20 -6.841823E-04 -9.083150E-07 9.017970E-06 27 G 1.314148E-17 0.0 -1.123426E+00 1.385172E-05 7.585519E-02 -1.074977E-18 0.0 -8.038118E-04 -1.009948E-06 9.012615E-06 28 G -1.091898E-17 -1.884208E-17 -1.123307E+00 8.928183E-06 7.585055E-02 8.859232E-19 1.543538E-18 -8.124351E-04 -6.510325E-07 9.350641E-06 29 G 4.758124E-18 -2.943992E-17 -1.123197E+00 1.286549E-05 7.585435E-02 -3.786244E-19 2.405971E-18 -8.205214E-04 -9.380911E-07 9.073568E-06 45 G 0.0 0.0 -3.725961E-01 0.0 5.863066E-02 0.0 0.0 3.070328E-02 0.0 -4.744958E-03 49 S -1.269181E-03 6.376013E-03 COMPLEX EIGENVALUE = -9.602517E-01, 1.025234E+04 C O M P L E X E I G E N V E C T O R NO. 22 (REAL/IMAGINARY) POINT ID.

TYPE 1

NO.

T1

T1

T2 -9.719831E-25 -1.074433E-26 -1.274938E-24 -1.404317E-26 8.587767E-23 -1.072967E-26 1.775076E-21 -1.117895E-26 3.166509E-21 -3.370484E-25 2.715861E-21 -1.072868E-24 1.435584E-20 -3.910426E-24 1.588413E-20 -3.020337E-24 1.900572E-20 -1.575166E-24 1.882168E-20 -1.938338E-24 1.891675E-20 -1.710528E-24 1.891675E-20 -1.710528E-24 -2.167423E-19 -1.453294E-21 1.165725E-19 2.653666E-21 -1.599295E-19 -2.792360E-21 1.891675E-20 -1.710528E-24 1.177647E-19 6.481823E-22 2.735508E-20 -2.146415E-24 -1.839683E-19

T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -5.420171E-19 -4.887460E-21 -8.214385E-19 -7.765938E-21 0.0 0.0 -6.261747E-20 1.396601E-22 -7.575990E-20 1.368252E-22 0.0

R1 4.008538E-03 7.587038E-05 -4.622021E-03 -7.379571E-05 -5.050509E-03 -9.960841E-05 2.585498E-03 1.063404E-05 8.187695E-03 1.249938E-04 2.645007E-03 1.042487E-04 -1.055022E-02 -3.926582E-05 -1.717034E-02 -1.474404E-04 -6.482240E-03 -8.404672E-05 1.796136E-02 1.266311E-04 4.767826E-03 1.068928E-05 4.767826E-03 1.068928E-05 1.144542E+00 8.549289E-06 1.144549E+00 8.563889E-06 1.144556E+00 8.577567E-06 4.767826E-03 1.068928E-05 4.773135E-03 1.069952E-05 4.782110E-03 1.071684E-05 -1.135005E+00

R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8.869857E-07 1.710962E-09 5.711277E-07 1.100592E-09 8.233890E-07 1.587523E-09 0.0 0.0 9.745244E-07 1.880641E-09 7.971618E-07 1.537178E-09 8.869637E-07

R3 6.837322E-04 1.157937E-05 3.586475E-04 6.774487E-06 -3.065130E-04 -3.581556E-06 -5.740994E-04 -9.334211E-06 -4.473000E-05 -4.002040E-06 7.411173E-04 6.619448E-06 8.278431E-04 1.037617E-05 -8.942012E-05 2.213668E-06 -1.283759E-03 -1.027467E-05 -1.802282E-03 -1.593006E-05 -1.672850E-03 -1.451660E-05 -1.672850E-03 -1.451660E-05 7.598489E-02 -1.427426E-07 7.598518E-02 -1.421725E-07 7.598494E-02 -1.426444E-07 7.598489E-02 -1.427398E-07 7.598489E-02 -1.427399E-07 7.598489E-02 -1.427400E-07 7.598489E-02

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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28

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-2.261795E-21 6.296017E-20 1.863909E-21 6.506194E-20 -7.998630E-22 0.0 0.0 7.765774E-02 1.437386E-05

0.0 2.920202E-19 3.244541E-21 3.850615E-19 5.057949E-21 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

1.283148E-05 -1.134997E+00 1.284609E-05 -1.134990E+00 1.285977E-05 6.379164E-03 7.781174E-05

1.711251E-09 5.711054E-07 1.100884E-09 8.233665E-07 1.587817E-09 0.0 0.0

-1.427371E-07 7.598460E-02 -1.433073E-07 7.598484E-02 -1.428356E-07 -4.055124E-04 -8.101913E-06

0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

Listing 7-43. Output for Servoelastic Stability Analysis of a Missile

7.13 Aeroservoelastic Stability Analysis of a Missile (Example HA145J) In this example, the aerodynamic forces are simply added to the previous Example HA110A . The additional Bulk Data entries beyond those required to solve the servoelastic stability problem provide the aerodynamic, spline, and flutter data. The input files MISSILE_STRUCT.DAT and MISSILE_SERVO.DAT (see Listing 7-6 and Listing 7-7) are INCLUDEd as before. The AERO entry gives the reference chord of 30.0 in., the reference density at sea level of 1.1468E-07 lb-sec2/in4, and specifies symmetry. CAERO1 101 divides the flipper semispan (see Figure 7-20) into 10 equal width strips and 15 equal chordwise segments on each strip to satisfy the guidelines of The Interpolation from Structural to Aerodynamic Models that the box chord Δx ≤ 0.08V/f where V is the minimum velocity of interest (500 ft/s) and f is the maximum frequency of interest (251 Hz) and that the box aspect ratios be less than 3. The CAERO1 continuation lists the coordinates of the leading edge at the root and tip and the root and tip chord lengths. PAERO1 1 defines the fuselage as the associated body that is described on CAERO2 and PAERO2 entries. CAERO2 1500 specifies the body characteristics on PAERO2 1520, the fractional divisions of the body on AEFACT 1505, the fractional divisions of the interference body on AEFACT 1506, and the body nose coordinates and length on the continuation entry. PAERO2 1520 provides data on the motion (in the z-direction for vertical motion), the reference half-width of the body (a representative value is recommended in the region of the lifting surface root), the aspect ratio (height/width) of the cross section approximated as an ellipse (AR = 1.0 in this example with a circular cross section), and AEFACT 1515 for the body half-widths at the end points given on AEFACT 1505. The PAERO2 field LRIB is left blank following the recommendation in Aerodynamic Modeling. The PAERO2 entry also specifies the θ1 - and θ2 -arrays for averaging the downwash on the interference body that is induced by the lifting surface bound vortices and doublets in the approximation to satisfy the no-flow condition through the body surface (only a θ1 -array is specified here on AEFACT 1518 at θ = 45°, 135°, 225°, and 315°, which are angles spaced midway between adjacent flippers in the cruciform configuration. The PAERO2 continuation specifies that all the interference elements use the θ1 -array. Linear spline entries are used to connect the aerodynamics and the structure along the hinge line at the midchord of the flipper and along the fuselage centerline. SPLINE2 1 connects the 150 aerodynamic boxes (numbers 101 through 250) to GRIDs 21 through 23 along the leading edge and GRIDs 27 through 29 along the trailing edge on the SET1 1 entry. The axis of the spline along the hinge line is the y-axis of CORD2R 1. The spline attachment flexibility is not used so DZ = 0.0, and no rotational connection to the flipper grid points is considered so DTHX = DTHY = -1.0; since DZ = 0.0, DTOR is arbitrarily set to 1.0. SPLlNE2 2 connects the six body aerodynamic elements (numbers 1500 through 1505) to GRIDs 1 through 10 on SET1 2. The attachment flexibilities are DZ = 0.0 and DTHY = 0.0 for both displacement and rotational spline connections, DTOR = 1.0 (always for bodies). DTHX is not used for bodies. For the vibration analysis, the modified Householder method (MHOU) is requested on the EIGR entry as before, and six vibration modes are again used in the modal formulation, PARAM,LMODES,6. PARAM,OPPHlPA,1, along with a DISP command in the Case Control Section, specifies the eigenvectors at both the structural and aerodynamic grid points. The values of Mach number and reduced frequency that provide the basis for interpolation are specified on MKAEROi entries. In this case, the MKAERO1 entry is selected for one Mach

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number, m = 0.8, and six reduced frequencies, k = 0.001, 0.1, 0.2, 0.5, 1.0, and 4.0, are specified to cover the range required by the velocities and frequencies of interest. The flutter data entries include FLFACT 1, which gives the density ratio of 1.0; FLFACT 2, which gives the Mach number 0.8; and FLFACT 3, which lists six velocities for the aeroservoelastic stability analysis ranging from 6000 to 12000 in/s. The first velocity is input as negative to obtain eigenvectors in addition to eigenvalues in the PK-method of solution. FLUTTER 40 specifies the PK-method of flutter analysis, refers to the FLFACT entries, and requests output for 16 modes. Sixteen roots are requested because the differentiation of the servo equations added six zero roots to the system of two rigid and three flexible body roots, two actuator/potentiometer roots, the gyro root, and the artificial actuator/flipper transfer function root. All real root data are output first. The last flutter data entry is PARAM,VREF to convert the output velocity units from in/s to ft/s by dividing by 12.0. The ENDDATA entry completes the Bulk Data Section.

Case Control Section The Case Control Section is changed for the flutter analysis from what was required in the servoelastic analysis. New title commands are used. The FMETHOD command replaces the CMETHOD command when the PK-method is specified for flutter analysis. Specifying SET1 = 1 through 2000, which includes all aerodynamic degrees of freedom, along with DISP requests modal displacements. The last command in the Case Control Section is BEGIN BULK. The Executive Control Section starts with the identification ID NXN, HA145J. It then specifies TIME 5 for 30.0 minutes of CPU time and SOL 145 for the Superelement Aerodynamic Flutter DMAP sequence. The CEND statement completes the Executive Control Section.

Output The input data echo is shown below in Listing 7-44 followed by the sorted Bulk Data entries in Listing 7-45. Selected items in the printed output are shown in Listing 7-46. The significant results are discussed here. The first output shown is the result of the vibration analysis: the first six frequencies, the generalized masses, and stiffnesses. These are the same as shown in Listing 7-43 in the previous section (except for round-off differences between SOLs 110 and 145). The next output shown is a representative real eigenvector, the fourth mode that is the first fuselage bending mode, and includes the SPOINT, EPOINT, and all aerodynamic grid point displacements. Following the typical real eigenvector is a series of eigenvalues and modal eigenvectors (participation factors) for the servo and vibration modes that were requested by the negative velocity V = −6000 in/s = −500 ft/s. Shown are the two actuator modes, the short period mode, the gyro mode, and three vibration modes. Also shown, for all of the velocities, are the zero frequency roots that are not listed in the Flutter Summary tables. The aerodynamic forces in this example have very little effect on the servoelastic system. The roots (shown next in the Flutter Summary tables) are changed only slightly from the previous Example HA110A , except for the rigid body pitch mode, which has become the short period mode and appears as Point 6 in the Flutter Summary tables. The PK-method of flutter analysis provides a solution for the short period mode whereas the K-method does not. With the flipper locked (by using scalar spring CELAS2 1 shown in the Bulk Data of Listing 7-44; results are not listed) the short period of the missile is lightly damped and has a constant reduced frequency k = 0.0186 for the range of airspeed from V = 500 to 1000 ft/s. Its frequency increases from f = 1.183 to 2.362 Hz, and its damping is constant at g = 2ζ = 0.2708. The purpose of the servo system is to increase the short period damping substantially. A reasonably well damped short period throughout the speed range of interest is obtained by adjusting the gyro gain to Kg = 0.13 deg per deg/s, and this is the design consideration that determines Kg . In the Flutter Summary for Point 6, the damping and frequency at V = 500 ft/s

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are g = 1.945 and f = 0.961 Hz and at V = 1000 ft/s the damping has decreased to g = 1.316 and the frequency has increased to f = 5.821 Hz. The short period characteristics in Point 6 of the Flutter Summary are the data of primary interest in this analysis. However, the other Points also deserve some comment. Point 7 is the gyro mode, which is unaffected by the airstream. Points 8 through 10 are the three fuselage modes and their frequencies are unaffected and their dampings are only slightly affected by the airstream. Point 11 is the artificial actuator/flipper transfer function mode, and the output indicating instability is meaningless. The first four Points correspond to some of the six zero frequency roots introduced by the differentiations required to avoid a singular mass matrix. Finally, Point 5 is the rigid body plunge mode that should also have a zero frequency, even in the presence of the airstream; the nonzero frequencies shown are round-off errors that result from the NX Nastran formulation using plunge and pitch (h, α) degrees of freedom rather than pitch and pitch rate (α, q) degrees of freedom in the equations of motion. The positive dampings shown should not be interpreted as instabilities. The data following the Flutter Summary tables are the complex eigenvectors of grid point deflections that were also generated by the negative velocity in the Bulk Data Section and the DISPlacement command in the Case Control Section. These correspond to the same servo and vibration modes for which the modal participation factors (discussed above) are shown. N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA145J $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA145J $$$$$$$$ $ $ $ MODEL DESCRIPTION TYPICAL AIR-TO-AIR MISSILE $ $ BAR MODEL WITH 10 FUSELAGE STATIONS $ $ AND AN ACTIVE CONTROLS SYSTEM. $ $ $ $ SOLUTION PK FLUTTER ANALYSIS METHOD USING $ $ DOUBLET-LATTICE METHOD AERODYNAMICS $ $ ON THE FLIPPERS AND SLENDER BODY $ $ AERODYNAMICS ON THE BODY $ $ $ $ OUTPUT XY PLOTS OF THE V-G FLUTTER DATA $ $ $ $$$$$$$$ $$$$$$$$ TIME 30 $ TIME IN CPU MINUTES SOL 145 $ FLUTTER ANALYSIS CEND EXAMPLE HA145J: FLEXIBLE MISSILE WITH CONTROL SYSTEM PAGE 2 SYMMETRIC FLIPPER AND BODY MOTIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, PK FLUTTER ANALYSIS C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA145J: FLEXIBLE MISSILE WITH CONTROL SYSTEM 2 SUBTI = SYMMETRIC FLIPPER AND BODY MOTIONS, DOUBLET-LATTICE AERO 3 LABEL = HALF-SPAN MODEL, PK FLUTTER ANALYSIS 4 ECHO = BOTH 5 MPC = 50 6 SDAMP = 1 $ MODAL DAMPING 7 TFL = 1 $ SERVO TRANSFER FUNCTION 8 METHOD = 14 $ MODIFIED HOUSEHOLDER REAL EIGENVALUE EXTRACTION METHOD 9 FMETHOD = 40 $ PK-FLUTTER METHOD 10 SET 1 = 1 THRU 2000 11 DISP = 1 12 OUTPUT(XYOUT) 13 CSCALE 2.0 14 PLOTTER NASTRAN 15 CURVELINESYMBOL = -6 16 YTTITLE = DAMPING G 17 YBTITLE = FREQUENCY F HZ 18 XTITLE = VELOCITY V (FT/SEC) 19 XTGRID LINES = YES 20 XBGRID LINES = YES 21 YTGRID LINES = YES 22 YBGRID LINES = YES 23 UPPER TICS = -1 24 TRIGHT TICS = -1 25 BRIGHT TICS = -1 26 XYPLOT VG / 1(G,F) 2(G,F) 3(G,F) 4(G,F) 5(G,F) 6(G,F) 7(G,F) 27 BEGIN BULK EXAMPLE HA145J: FLEXIBLE MISSILE WITH CONTROL SYSTEM PAGE 3 SYMMETRIC FLIPPER AND BODY MOTIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, PK FLUTTER ANALYSIS I N P U T B U L K D A T A D E C K E C H O .

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Flutter Analysis Sample Problems

$*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS SPECIFIC $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DEFINITIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ INCLUDE MISSILE_STRUCT.DAT $ $ SPRING FOR LOCKING FLIPPER ROTATION $ $ EID K G1 C1 G2 C2 GE S $CELAS2 1 1.0+10 12 5 24 5 $ INCLUDE MISSILE_SERVO.DAT $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ $ REFERENCE LENGTHS PLUS SYMMETRY KEYS. SYMXZ = 0 INDICATES $ $ THAT THE MODEL IS MOUNTED WITH NO ROOT REFLECTION PLANE; $ $ SYMXY = 0 INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH $ $ FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 30.0 1.1468-71 $ $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET LATTICE AERODYNAMICS. $ $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS. $ $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ $ FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS $ $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY $ $ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ $ EXTRA POINT IDS. $ $ $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID +CONT CAERO1 101 1 10 15 1 +CA1 $ ( FWD LEFT POINT ) ROOTCHORD ( FWD RIGHT POINT ) TIP CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CA1 60.0 10.0 0.0 30.0 60.0 30.0 0.0 30.0 $ $ $ THE PAERO1 ENTRY DEFINES CAERO2 ENTRIES THAT DEFINE ASSOCIATED $ $ BODIES. $ $ $ $ PID B1 B2 B3 ETC PAERO1 1 1500 $ $ $ THE CAERO2 ENTRY DEFINES AN AERODYNAMIC BODY FOR DOUBLET$ $ LATTICE AERODYNAMICS. IT LISTS ITS PAERO2 ENTRY, THE $ $ COORDINATE SYSTEM FOR LOCATING POINT 1 AT THE NOSE OF THE $ $ BODY, THE NUMBER OF UNIFORM SLENDER BODY AND INTERFERENCE $ $ ELEMENTS, OR IF NON-UNIFORM IT IDENTIFIES AEFACT ENTRIES ON $ $ WHICH THE ELEMENTS ARE DEFINED, AND IT IDENTIFIES THE $ $ ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY LISTS $ $ THE COORDINATES OF THE NOSE POINT AND THE LENGTH OF THE BODY. $ $ $ $ EID PID CP NSB NINT LSB LINT IGID +CONT CAERO2 1500 1520 1505 1506 1 +CA2 $ X1 Y1 Z1 X12 +CA2 -60.0 0.0 0.0 150.0 $ $ $ SID D1 D2 D3 ETC AEFACT 1505 0.0 0.1 0.2 0.4 0.6 0.8 1.0 AEFACT 1506 .6 .8 1. $ $ THE PAERO2 ENTRY DEFINES THE CROSS-SECTION PROPERTIES OF $ $ AERODYNAMIC BODIES. IT LISTS THE ORIENTATION OF MOTION $ $ OF THE BODY, ITS HALF-WIDTH AND ASPECT RATIO (HEIGHT/ $ $ WIDTH), IDENTIFIES AEFACT ENTRIES FOR ADDITIONAL DATA AND $ $ IDENTIFIES THE FIRST AND LAST INTERFERENCE ELEMENTS OF THE $ $ BODY TO USE THE THETA ARRAYS DESIGNATED BY LTH1 AND LTH2. $ $ $ $ PID ORIENT WIDTH AR LRSB LRIB LTH1 LTH2 PAERO2 1520 Z 10.0 1.0 1515 1518 +PA2 $ THI1 THN1 THI2 THN2 +PA2 1 2 $ $ $ (LRSB) SLENDER BODY HALF-WIDTHS $ $ $ $ SID D1 D2 D3 D4 D5 D6 D7 AEFACT 1515 0.0 10.0 10.0 10.0 10.0 10.0 10.0 $ $ $ (LTH1 AND 2) THETA ARRAYS $ $ $ $ THETA IS A POSITIVE ROTATION ABOUT THE X-AXIS, SO 0. LIES $

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

$ $ $ AEFACT $ $ $ $ $ $ $ $ $ $ $ $ SPLINE2 $ +SP1 $ $ $ $ $ SET1 $ $ $ $ $ $ $ CORD2R $ +CR1 $ $ $ $ $ $ $ $ $ $ SPLINE2 $ +SP2 $ $ SET1 $

AT THE JUNCTURE OF THE FLIPPER AND THE BODY. SID 1518

D1 45.

D2 135.

D4 315.

ETC

THE SPLINE2 ENTRY SPECIFIES A BEAM SPLINE FOR INTERPOLATION OVER THE REGION OF THE CAERO ENTRY (ID1 AND ID2 ARE THE FIRST AND LAST BOXES IN THIS REGION). SETG REFERS TO A SET1 ENTRY WHERE THE STRUCTURAL GRID POINTS ARE DEFINED. DZ AND DTOR ARE SMOOTHING CONSTANTS FOR LINEAR ATTACHMENT AND TORSIONAL FLEXIBILITIES. DTHX AND DTHY ARE ROTATIONAL ATTACHMENT FLEXIBILITIES. CID IDENTIFIES THE CORD2R ENTRY THAT DEFINES THE SPLINE AXIS. EID 1 DTHX -1.0

CAERO 101 DTHY -1.0

ID1 101

ID2 250

SETG 1

DZ 0.0

DTOR 1.0

CID 1

THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. SID 1

G1 21

G2 22

G3 23

ETC 27

28

CID 1 C1 100.

RID

A1 75.0 C3

C2

A2

A3

B1 75.0

B2

B3 30.

THE NEXT ENTRY AND THE SET1 ENTRY IT SPECIFIES DEFINE A BEAM SPLINE TO INTERPOLATE BETWEEN ELEMENTS 101 AND 109 ON THE FUSELAGE. CID DEFAULTS TO THE BASIC COORDINATE SYSTEM WHICH DEFINES THE Y AXIS OF THE SPLINE AS COINCIDENT WITH THE BASIC Y AXIS. DZ, DTOR, DTHX AND DTHY ARE ATTACHMENT FLEXIBILITIES WHICH, FOR BODIES, ARE TAKEN AS DTOR = 1.0 AND DTHX IS NOT USED. EID 2 DTHX

CAERO 1500 DTHY 0.0

ID1 1500

ID2 1505

SETG 2

SID 2

G1 1

G2 THRU

G2 10

ETC

DZ 0.0

$ $ $ $ $ $ $ $ $ $ $ +SP1 $ $ $ $

29

THE CORD2R ENTRY DEFINES THE COORDINATE SYSTEM IN WHICH THE BEAM SPLINE EXTENDS ALONG THE Y-AXIS. IT LISTS THE ORIGIN, A POINT ALONG THE Z-AXIS AND A POINT IN THE X-Z PLANE. IN THIS CASE, THE SPLINE AXIS IS ALIGNED WITH THE HINGE LINE.

DTOR 1.0

$ $ $ $ $ $ +CR1 $ $ $ $ $ $ $ $ $

CID +SP2 $ $

$ $ $ $ EIGR $+EIG +HOU $ $ $ $ $ $ $ $ $ $ PARAM $ $ $ $ $ $ $ PARAM $ $ $ $ $ $ $ $ $ $ MKAERO1 $ +MKA

D3 225.

$ $

$ * * VIBRATION ANALYSIS * * SID 14 NORM MAX

METHOD MHOU G

F1

F2

NE

ND 6

C * VIBRATION MODES *

THE PARAM, OPPHIPA = 1 (WHEN ACCOMPANIED BY A DISPLACEMENT REQUEST IN THE CASE CONTROL DECK), WILL GENERATE THE VIBRATION MODE DISPLACEMENTS FOR BOTH THE STRUCTURAL GRID POINTS AND THE AERODYNAMIC GRID POINTS (BOX CENTERLINE MIDPOINTS) IN THE OUTPUT. N V1 OPPHIPA 1

THE PARAM,LMODES,N ENTRY SPECIFIES THAT N VIBRATION MODES ARE TO BE USED IN THE ANALYSIS. V1 6

V2 * * AERODYNAMIC CONDITIONS * *

ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED TO GENERATE GENERALIZED AERO FORCE MATRICES. IF MORE THAN EIGHT MACH NO.S OR REDUCED FREQUENCIES ARE REQUIRED A SECOND MKAERO1 IS NECESSARY. M1 0.8 K1 0.001

$ $ $ $ $ $ $ $ $

V2

* * * AEROSERVOELASTIC STABILITY ANALYSIS * * *

N LMODES

$ $ +EIG +HOU

M2

M3

ETC

K2 0.1

K3 0.2

K4 0.5

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ +MKA

K5 1.0

ETC

NX Nastran Aeroelastic Analysis User’s Guide

7-167

Chapter 7

Flutter Analysis Sample Problems

$ $ * * FLUTTER SOLUTION PARAMETERS * * $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE (DEFAULT IS 10-3). $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 40 PK 1 2 3 L 16 0.01 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ NEGATIVE VELOCITIES ARE CODES TO COMPUTE AND PRINT EIGEN$ VECTORS. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 1.0 FLFACT 2 0.8 FLFACT 3 -6000. 7200. 8400. 9600. 10800. 12000. $ $ THE PARAM,VREF,C ENTRY SPECIFIES A CONVERSION FACTOR TO BE $ USED TO CONVERT THE DIMENSIONS OF THE OUTPUT VELOCITIES BY $ DIVIDING BY C, IN THIS CASE BY 12.0 IN/FT TO PRINT VEL$ OCITIES IN FT/SEC RATHER THAN IN/SEC. $ $ N V1 V2 PARAM VREF 12.0 $ ENDDATA INPUT BULK DATA CARD COUNT = 465

7-168

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $

Flutter Analysis Sample Problems

Listing 7-44. Input Files for Aeroservoelastic Stability Analysis of a Missile EXAMPLE HA145J: FLEXIBLE MISSILE WITH CONTROL SYSTEM PAGE 12 SYMMETRIC FLIPPER AND BODY MOTIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, PK FLUTTER ANALYSIS S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 1505 0.0 0.1 0.2 0.4 0.6 0.8 1.0 2AEFACT 1506 .6 .8 1. 3AEFACT 1515 0.0 10.0 10.0 10.0 10.0 10.0 10.0 4AEFACT 1518 45. 135. 225. 315. 5AERO 30.0 1.1468-71 6CAERO1 101 1 10 15 1 +CA1 7+CA1 60.0 10.0 0.0 30.0 60.0 30.0 0.0 30.0 8CAERO2 1500 1520 1505 1506 1 +CA2 9+CA2 -60.0 0.0 0.0 150.0 10CBAR 1 1 1 2 0.0 0.0 1.0 11CBAR 2 1 2 3 0.0 0.0 1.0 12CBAR 3 1 3 4 0.0 0.0 1.0 13CBAR 4 1 4 45 0.0 0.0 1.0 14CBAR 5 1 5 6 0.0 0.0 1.0 15CBAR 6 1 6 7 0.0 0.0 1.0 16CBAR 7 1 7 8 0.0 0.0 1.0 17CBAR 8 1 8 9 0.0 0.0 1.0 18CBAR 9 1 9 11 0.0 0.0 1.0 19CBAR 10 1 11 10 0.0 0.0 1.0 20CBAR 45 1 45 5 0.0 0.0 1.0 21CONM2 1 1 50. 22CONM2 2 2 50. 23CONM2 3 3 50. 24CONM2 4 4 50. 25CONM2 5 5 50. 26CONM2 6 6 50. 27CONM2 7 7 50. 28CONM2 8 8 50. 29CONM2 9 9 52.833 30CONM2 10 10 52.167 31CORD2R 1 75.0 75.0 30. +CR1 32+CR1 100. 33CQUAD4 1 1 21 22 25 24 34CQUAD4 2 1 22 23 26 25 35CQUAD4 3 2 24 25 28 27 36CQUAD4 4 2 25 26 29 28 37EIGR 14 MHOU 6 +HOU 38+HOU MAX 39EPOINT 50 51 52 53 54 40FLFACT 1 1.0 41FLFACT 2 0.8 42FLFACT 3 -6000. 7200. 8400. 9600. 10800. 12000. 43FLUTTER 40 PK 1 2 3 L 16 0.01 44GRID 1 -52.5 0. 0. 246 45GRID 2 -37.5 0. 0. 246 46GRID 3 -22.5 0. 0. 246 47GRID 4 -7.5 0. 0. 246 48GRID 5 7.5 0. 0. 246 49GRID 6 22.5 0. 0. 246 50GRID 7 37.5 0. 0. 246 51GRID 8 52.5 0. 0. 246 52GRID 9 67.5 0. 0. 246 53GRID 10 82.5 0. 0. 246 54GRID 11 75.0 0. 0. 246 55GRID 12 75.0 10. 0. 56GRID 21 60.0 10. 0. 26 57GRID 22 60.0 20. 0. 6 58GRID 23 60.0 30. 0. 6 59GRID 24 75.0 10. 0. 6 60GRID 25 75.0 20. 0. 6 61GRID 26 75.0 30. 0. 6 62GRID 27 90.0 10. 0. 26 63GRID 28 90.0 20. 0. 6 64GRID 29 90.0 30. 0. 6 65GRID 45 0.0 0. 0. 1246 66MAT1 1 1.0+7 0.33 0.0 67MAT1 10 1.0+10 0.30 .0188889 68MAT1 20 1.0+10 0.30 .0144444 69MKAERO1 0.8 +MKA 70+MKA 0.001 0.1 0.2 0.5 1.0 4.0 71MPC 50 24 1 1.0 12 1 -1.0 72MPC 50 24 2 1.0 12 2 -1.0 73MPC 50 24 3 1.0 12 3 -1.0 74MPC 50 24 4 1.0 12 4 -1.0 75MPC 50 49 1.0 24 5 -1.0 +MPC50 76+MPC50 12 5 1.0 77PAERO1 1 1500 78PAERO2 1520 Z 10.0 1.0 1515 1518 +PA2 79+PA2 1 2 80PARAM COUPMASS1 81PARAM GRDPNT 45 82PARAM LMODES 6 83PARAM OPPHIPA 1 84PARAM VREF 12.0 85PARAM WTMASS .0025907 7-169 NX Nastran Aeroelastic Analysis User’s Guide 86PBAR 1 1 1.0+8 67.655 67.655 1.0+8 87PSHELL 1 10 .500 10 88PSHELL 2 20 .500 20

Flutter Analysis Sample Problems

Chapter 7

919293949596979899100101102103104105106107108109110111112-

SET1 SPLINE2 +SP1 SPLINE2 +SP2 SPOINT TABDMP1 +DMP1 +DMP2 TF +4950 TF +5053 TF +511 TF +550 TF +531 +532 TF +5445 ENDDATA TOTAL COUNT=

2 1 -1.0 2 49 1 0.0 1000.0 1 50 1 53 1 54 1 50 1 51 52 1 45

1 101 -1.0 1500 0.0

THRU 101

10 250

1

0.0

1.0

1500

1505

2

0.0

1.0

0.03 0.08 49

45.0 ENDT

0.03

125.4

0.05

248.2

1

+SP1 +SP2

0.08

1.0+8

+DMP1 +DMP2 +4950

-1.0+8 50

1.0

0.002

+5053

1.0

+511

1.0

+550

1.0

+531 +532

-50.0 51 -1.0 52 -1.0 53 54 5

1.0 -0.13

-1.0 1.0 3.7136-37.0362-6

+5445

113

Listing 7-45. Sorted Bulk Data Entries for Aeroservoelastic Stability Analysis of a Missile EXAMPLE HA145J: FLEXIBLE MISSILE WITH CONTROL SYSTEM SYMMETRIC FLIPPER AND BODY MOTIONS, DOUBLET-LATTICE AERO HALF-SPAN MODEL, PK FLUTTER ANALYSIS R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED NO. ORDER MASS 1 5 7.130438E-10 2.670288E-05 4.249895E-06 4.623947E-01 2 3 2.192974E-08 1.480869E-04 2.356877E-05 1.776271E-02 3 2 5.170114E-07 7.190350E-04 1.144380E-04 4.765784E-03 4 1 8.055728E+04 2.838261E+02 4.517233E+01 5.257575E-01 5 4 6.341577E+05 7.963402E+02 1.267415E+02 7.764813E-01 6 6 2.487909E+06 1.577311E+03 2.510368E+02 7.707402E-01 EIGENVALUE = 8.055728E+04 CYCLES = 4.517233E+01 R E A L E I G E N V E C T O R N O . POINT ID.

7-170

TYPE 1 2 3 4 5 6 7 8 9 10 11 12 21 22 23 24 25 26 27 28 29 45 49 50 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

T1 G G G G G G G G G G G G G G G G G G G G G G S E G G G G G G G G G G G G G G G G G G G G G G

T2 -5.504608E-24 -7.194693E-24 -5.496885E-24 -5.726971E-24 -3.209346E-22 -9.569880E-22 -1.609433E-21 -1.367960E-21 7.585179E-21 7.106629E-21 7.386534E-21 7.386534E-21 -1.325798E-18 2.458886E-18 -2.585523E-18 7.386534E-21 5.770939E-19 5.896732E-21 -2.104592E-18 1.721998E-18 -7.449252E-19 0.0 2.979981E-02 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.446147E-18 -7.068179E-18 0.0 1.530867E-19 1.675153E-19 0.0 2.979970E-18 4.652544E-18 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

R1 1.000000E+00 4.148082E-01 -1.164419E-01 -5.268358E-01 -7.543173E-01 -7.618728E-01 -5.486975E-01 -1.505693E-01 3.706050E-01 9.466017E-01 6.554550E-01 6.554550E-01 5.243307E-01 5.245903E-01 5.248336E-01 6.554550E-01 6.556371E-01 6.559448E-01 7.866180E-01 7.868776E-01 7.871209E-01 -6.664273E-01 0.0 5.330999E-01 5.505857E-01 5.680715E-01 5.855574E-01 6.030432E-01 6.205291E-01 6.380149E-01 6.555007E-01 6.729865E-01 6.904723E-01 7.079582E-01 7.254440E-01 7.429298E-01 7.604156E-01 7.779014E-01 5.331525E-01 5.506384E-01 5.681242E-01 5.856100E-01 6.030958E-01 6.205816E-01 6.380675E-01

PAGE

21

GENERALIZED STIFFNESS 3.297077E-10 3.895316E-10 2.463964E-09 4.235359E+04 4.924116E+05 1.917531E+06 4

R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.041461E-05 1.958227E-05 2.823276E-05 0.0 3.341751E-05 2.733380E-05 3.041434E-05 1.958200E-05 2.823249E-05 0.0

R3 3.957338E-02 3.789160E-02 3.214863E-02 2.184270E-02 8.055649E-03 -7.057895E-03 -2.094803E-02 -3.141067E-02 -3.726979E-02 -3.895797E-02 -3.854272E-02 -3.854272E-02 -8.743007E-03 -8.732853E-03 -8.741221E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742815E-03 -8.752967E-03 -8.744600E-03 1.527356E-02

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140

G

G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

6.555533E-01 6.730391E-01 6.905249E-01 7.080108E-01 7.254966E-01 7.429824E-01 7.604682E-01 7.779540E-01 5.332049E-01 5.506907E-01 5.681765E-01 5.856624E-01 6.031482E-01 6.206340E-01 6.381198E-01 6.556056E-01 6.730915E-01 6.905773E-01 0.0 7.080631E-01 7.255489E-01 7.430347E-01 7.605206E-01 7.780064E-01 5.332567E-01 5.507426E-01 5.682284E-01 5.857142E-01 6.032000E-01 6.206858E-01 6.381717E-01 6.556575E-01 6.731433E-01 6.906291E-01 7.081149E-01 7.256008E-01 7.430866E-01 7.605724E-01 7.780582E-01 5.333079E-01 5.507937E-01 5.682795E-01 5.857654E-01 6.032512E-01 6.207370E-01 6.382228E-01 6.557087E-01 6.731945E-01 6.906803E-01 7.081661E-01 7.256519E-01 7.431378E-01 7.606236E-01 7.781094E-01 5.333582E-01 5.508440E-01 5.683298E-01 5.858157E-01 6.033015E-01 6.207873E-01 6.382731E-01 6.557589E-01 6.732448E-01 6.907306E-01 7.082164E-01 7.257022E-01 7.431880E-01 7.606739E-01 7.781597E-01 5.334076E-01 5.508934E-01 5.683792E-01 5.858651E-01 6.033509E-01 6.208367E-01 6.383225E-01 6.558083E-01 6.732942E-01 6.907800E-01 7.082658E-01 7.257516E-01 7.432374E-01 7.607233E-01 7.782091E-01 5.334563E-01 5.509422E-01 5.684280E-01 5.859138E-01 6.033996E-01 6.208854E-01 6.383713E-01 6.558571E-01

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 -8.742911E-03 0.0 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03 0.0 -8.742911E-03

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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214 215 216 217 218 219 220

G

G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 1500 1501 1502 1503 1504 1505

EIGENVALUE = EIGENVECTOR

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-4.41464E+02

0.00000E+00

-2.07264E-03 -2.35549E-01 1.00000E+00 3.07997E-04 -8.04570E-04 -4.60887E-04 7.71414E-02 -2.59784E-03 7.71413E-02 -7.97392E-02 -2.59784E-03

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

-4.70153E+01

0.00000E+00

6.733429E-01 0.0 -8.742911E-03 6.908287E-01 0.0 -8.742911E-03 7.083145E-01 0.0 -8.742911E-03 7.258004E-01 0.0 -8.742911E-03 7.432862E-01 0.0 -8.742911E-03 7.607720E-01 0.0 -8.742911E-03 7.782578E-01 0.0 -8.742911E-03 0.0 5.335046E-01 0.0 -8.742911E-03 5.509904E-01 0.0 -8.742911E-03 5.684762E-01 0.0 -8.742911E-03 5.859621E-01 0.0 -8.742911E-03 6.034479E-01 0.0 -8.742911E-03 6.209337E-01 0.0 -8.742911E-03 6.384195E-01 0.0 -8.742911E-03 6.559053E-01 0.0 -8.742911E-03 6.733912E-01 0.0 -8.742911E-03 6.908770E-01 0.0 -8.742911E-03 7.083628E-01 0.0 -8.742911E-03 7.258486E-01 0.0 -8.742911E-03 7.433344E-01 0.0 -8.742911E-03 7.608203E-01 0.0 -8.742911E-03 7.783061E-01 0.0 -8.742911E-03 5.335525E-01 0.0 -8.742911E-03 5.510383E-01 0.0 -8.742911E-03 5.685242E-01 0.0 -8.742911E-03 5.860100E-01 0.0 -8.742911E-03 6.034958E-01 0.0 -8.742911E-03 6.209816E-01 0.0 -8.742911E-03 6.384674E-01 0.0 -8.742911E-03 6.559533E-01 0.0 -8.742911E-03 6.734391E-01 0.0 -8.742911E-03 6.909249E-01 0.0 -8.742911E-03 7.084107E-01 0.0 -8.742911E-03 7.258965E-01 0.0 -8.742911E-03 7.433824E-01 0.0 -8.742911E-03 7.608682E-01 0.0 -8.742911E-03 7.783540E-01 0.0 -8.742911E-03 1.000000E+00 0.0 3.957338E-02 4.148082E-01 0.0 3.789160E-02 -3.409625E-01 0.0 2.754156E-02 -7.864329E-01 0.0 5.061143E-04 -3.692509E-01 0.0 -2.672314E-02 6.554380E-01 0.0 -3.854272E-02 EIGENVECTOR FROM THE PK METHOD

EIGENVECTOR FROM THE PK METHOD EIGENVALUE = EIGENVECTOR

2.06083E-02 0.00000E+00 3.21790E-01 0.00000E+00 1.00000E+00 0.00000E+00 2.58993E-04 0.00000E+00 1.54166E-04 0.00000E+00 7.22003E-05 0.00000E+00 4.72282E-02 0.00000E+00 6.98930E-03 0.00000E+00 4.72282E-02 0.00000E+00 -4.02389E-02 0.00000E+00 6.98929E-03 0.00000E+00 A ZERO FREQUENCY ROOT HAS EMERGED. WHEN THE MACH NO., DENSITY AND VELOCITY ARE COMPATIBLE IT MAY BE INTERPRETED TWO WAYS DEPENDING ON THE SIGN OF THE REAL PART: 1. (-) A MODE IS CRITICALLY DAMPED, OR, 2. (+) THE SYSTEM IS DIVERGING. ONLY THE MOST CRITICAL ( I.E., MOST POSITIVE REAL ROOTS ) ARE PRINTED IN THE FLUTTER SUMMARY. FOR INFORMATIONAL PURPOSES, THE REMAINING REAL ROOTS ARE PRINTED HERE. LESS CRITICAL REAL ROOTS FOR LOOP 1 MACH VELOCITY DENSITY 8.00000E-01 6.00000E+03 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -3.1845191E+00 -4.4146350E+02 0.0000000E+00 0.0000 -3.3914748E-01 -4.7015335E+01 0.0000000E+00 0.0000 -5.1336172E-03 -7.1166307E-01 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 EIGENVECTOR FROM THE PK METHOD EIGENVALUE = -5.87527E+00 6.04031E+00 EIGENVECTOR 5.10478E-02 -4.51669E-02 1.00000E+00 0.00000E+00 8.56611E-01 -4.07508E-02

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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

-2.87211E-05 -5.65440E-06 -6.89861E-07 1.84069E-03 1.85208E-03 1.84094E-03 1.02953E-05 1.85175E-03

-4.78540E-05 -7.08999E-06 -3.76186E-06 -1.92249E-03 -1.47831E-03 -1.92258E-03 4.43285E-04 -1.47868E-03

-2.66003E+02

2.69353E+02

-2.40905E-03 -2.40943E-01 1.00000E+00 -2.38888E-06 4.50874E-04 2.29312E-04 7.71210E-02 -3.38958E-01 7.71210E-02 -4.16079E-01 -3.38958E-01

7.67364E-04 9.66042E-03 0.00000E+00 -1.09465E-03 1.01734E-03 4.35329E-04 -7.53787E-04 -2.32821E-02 -7.53785E-04 -2.25283E-02 -2.32821E-02

-5.43000E+00

2.82303E+02

-3.64972E-03 -3.23286E-01 1.00000E+00 -3.28786E-01 1.12979E-03 4.51962E-04 7.15586E-02 -3.17011E-01 7.15585E-02 -3.88570E-01 -3.17011E-01

1.30136E-03 -4.67396E-02 0.00000E+00 8.42691E-01 -8.45627E-04 -1.82433E-04 2.76581E-02 3.31767E-01 2.76581E-02 3.04109E-01 3.31767E-01

-2.05338E+01

7.93753E+02

-3.79826E-03 -2.30486E-01 1.00000E+00 6.08011E-04 7.40922E-01 3.58124E-03 2.26712E-02 -4.78225E-01 2.26712E-02 -5.00896E-01 -4.78225E-01

-1.31558E-03 -1.15603E-02 0.00000E+00 -9.07446E-05 8.49715E-02 2.60128E-05 -5.44824E-03 4.64349E-01 -5.44824E-03 4.69797E-01 4.64349E-01

-6.64526E+01

1.56920E+03

-1.38204E-03 -2.17917E-01 1.00000E+00 5.71649E-03 -6.97955E-03 7.79734E-01 -3.05448E-03 1.58821E-01 -3.05447E-03 1.61876E-01 1.58821E-01

1.99750E-03 -3.24386E-03 0.00000E+00 1.40264E-03 -3.90675E-03 -5.81974E-02 6.18426E-03 -6.80447E-01 6.18426E-03 -6.86631E-01 -6.80447E-01

EIGENVECTOR FROM THE PK METHOD EIGENVALUE = EIGENVECTOR

EIGENVECTOR FROM THE PK METHOD EIGENVALUE = EIGENVECTOR

EIGENVECTOR FROM THE PK METHOD EIGENVALUE = EIGENVECTOR

EIGENVECTOR FROM THE PK METHOD EIGENVALUE = EIGENVECTOR

LESS CRITICAL REAL ROOTS FOR LOOP

2

MACH VELOCITY DENSITY 8.00000E-01 7.20000E+03 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -2.6566422E+00 -4.4194196E+02 0.0000000E+00 0.0000 -2.3825532E-01 -3.9634628E+01 0.0000000E+00 0.0000 -5.5663930E-03 -9.2598951E-01 0.0000000E+00 0.0000 -5.0779374E-07 -8.4473315E-05 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 LESS CRITICAL REAL ROOTS FOR LOOP 3 MACH VELOCITY DENSITY 8.00000E-01 8.40000E+03 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -2.2799907E+00 -4.4249875E+02 0.0000000E+00 0.0000 -4.8009757E-02 -9.3176956E+00 0.0000000E+00 0.0000 -5.3678605E-07 -1.0417900E-04 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 LESS CRITICAL REAL ROOTS FOR LOOP 4 MACH VELOCITY DENSITY 8.00000E-01 9.60000E+03 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -1.9978441E+00 -4.4313141E+02 0.0000000E+00 0.0000 -3.5784751E-02 -7.9372292E+00 0.0000000E+00 0.0000 -3.7959325E-03 -8.4195602E-01 0.0000000E+00 0.0000 -1.1922250E-05 -2.6444122E-03 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00

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5

MACH VELOCITY DENSITY 8.00000E-01 1.08000E+04 1.00000E+00 COMPLEX EIGENVALUE -4.4384097E+02 0.0000000E+00 -7.3752437E+00 0.0000000E+00 -8.1752992E-01 0.0000000E+00 -7.8921029E-03 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 6 MACH VELOCITY DENSITY 8.00000E-01 1.20000E+04 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -1.6036615E+00 -4.4462476E+02 0.0000000E+00 0.0000 -2.5603099E-02 -7.0986128E+00 0.0000000E+00 0.0000 -8.4731131E-05 -2.3492213E-02 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 0.0000000E+00 0.0000000E+00 0.0000000E+00 FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0000 9.9999996E+24 5.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 6.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 7.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 8.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 9.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 1.0000000E+03 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 POINT = 2 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0000 9.9999996E+24 5.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 6.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 7.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 8.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 9.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 1.0000000E+03 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 POINT = 3 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0000 9.9999996E+24 5.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 6.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 7.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 8.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 9.0000000E+02 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000 9.9999996E+24 1.0000000E+03 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 POINT = 4 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0000 9.9999996E+24 5.0000000E+02 4.6909195E-03 0.0000000E+00 6.5029275E-01 0.0000000E+00 0.0000 9.9999996E+24 6.0000000E+02 5.0777629E-07 0.0000000E+00 8.4470412E-05 0.0000000E+00 0.0000 9.9999996E+24 7.0000000E+02 5.3692264E-07 0.0000000E+00 1.0420552E-04 0.0000000E+00 0.0000 9.9999996E+24 8.0000000E+02 1.0879828E-05 0.0000000E+00 2.4131979E-03 0.0000000E+00 0.0000 9.9999996E+24 9.0000000E+02 3.6093188E-05 0.0000000E+00 9.0063466E-03 0.0000000E+00 0.0000 9.9999996E+24 1.0000000E+03 2.9758082E-04 0.0000000E+00 8.2506068E-02 0.0000000E+00 KFREQ DAMPING 0.0000 -1.7787051E+00 0.0000 -2.9556496E-02 0.0000 -3.2762741E-03 0.0000 -3.1627824E-05 0.0000 0.0000000E+00 0.0000 0.0000000E+00 LESS CRITICAL REAL ROOTS FOR LOOP

POINT = KFREQ 0.0000 0.0000 0.0007 0.0000 0.0000 0.0001 POINT = KFREQ 0.0151 0.0116 0.0188 0.0345 0.0415 0.0457 POINT = KFREQ 0.6734 0.5618 0.4821 0.4223 0.3758 0.3387 POINT = KFREQ 0.7058 0.5873 0.5028 0.4395 0.3903 0.3509 POINT = KFREQ 1.9844 1.6529 1.4161 1.2386 1.1005 0.9900

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5 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.2529122E+05 5.0000000E+02 4.4505945E-01 5.0811208E-04 7.1044004E-04 3.1925624E-03 9.9999996E+24 6.0000000E+02 5.0038975E-03 0.0000000E+00 8.3241630E-01 0.0000000E+00 1.5001816E+03 7.0000000E+02 5.3292379E-02 5.9410647E-02 9.9467058E-03 3.7328812E-01 9.9999996E+24 8.0000000E+02 3.4945598E-03 0.0000000E+00 7.7511013E-01 0.0000000E+00 9.9999996E+24 9.0000000E+02 3.0201059E-03 0.0000000E+00 7.5360817E-01 0.0000000E+00 6.8985298E+03 1.0000000E+03 -4.8341873E-01 1.8456679E-02 -2.8030246E-02 1.1596674E-01 6 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 6.6221802E+01 5.0000000E+02 -1.9453560E+00 9.6134466E-01 -5.8752737E+00 6.0403070E+00 8.5954094E+01 6.0000000E+02 -3.3938093E+00 8.8878101E-01 -9.4761534E+00 5.5843759E+00 5.3070381E+01 7.0000000E+02 -4.6146574E+00 1.6794069E+00 -2.4346989E+01 1.0552025E+01 2.9027264E+01 8.0000000E+02 -2.2388120E+00 3.5090857E+00 -2.4680927E+01 2.2048237E+01 2.4096344E+01 9.0000000E+02 -1.6415869E+00 4.7555580E+00 -2.4525351E+01 2.9880053E+01 2.1873892E+01 1.0000000E+03 -1.3160546E+00 5.8208189E+00 -2.4066219E+01 3.6573284E+01 7 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.4850420E+00 5.0000000E+02 -1.9751244E+00 4.2868805E+01 -2.6600250E+02 2.6935266E+02 1.7798563E+00 6.0000000E+02 -1.9701785E+00 4.2921646E+01 -2.6566345E+02 2.6968466E+02 2.0741448E+00 7.0000000E+02 -1.9667431E+00 4.2970367E+01 -2.6550125E+02 2.6999078E+02 2.3678150E+00 8.0000000E+02 -1.9644917E+00 4.3018208E+01 -2.6549258E+02 2.7029138E+02 2.6606598E+00 9.0000000E+02 -1.9629377E+00 4.3068848E+01 -2.6559485E+02 2.7060956E+02 2.9524899E+00 1.0000000E+03 -1.9618177E+00 4.3124268E+01 -2.6578488E+02 2.7095776E+02 8 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 1.4169151E+00 5.0000000E+02 -3.8469266E-02 4.4929989E+01 -5.4300036E+00 2.8230347E+02 1.7026080E+00 6.0000000E+02 -4.0494051E-02 4.4869030E+01 -5.7080503E+00 2.8192044E+02 1.9887151E+00 7.0000000E+02 -4.2430785E-02 4.4816254E+01 -5.9740176E+00 2.8158884E+02 2.2752347E+00 8.0000000E+02 -4.4417746E-02 4.4768635E+01 -6.2471261E+00 2.8128964E+02 2.5622046E+00 9.0000000E+02 -4.6537519E-02 4.4723808E+01 -6.5387073E+00 2.8100800E+02 2.8496542E+00 1.0000000E+03 -4.8829917E-02 4.4680496E+01 -6.8541541E+00 2.8073584E+02 9 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 5.0393486E-01 5.0000000E+02 -5.1738407E-02 1.2632977E+02 -2.0533770E+01 7.9375342E+02 6.0499519E-01 6.0000000E+02 -5.1661655E-02 1.2627268E+02 -2.0494041E+01 7.9339471E+02 7.0614982E-01 7.0000000E+02 -5.2128483E-02 1.2621509E+02 -2.0669800E+01 7.9303284E+02 8.0738616E-01 8.0000000E+02 -5.2975584E-02 1.2615916E+02 -2.0996380E+01 7.9268140E+02 9.0869176E-01 9.0000000E+02 -5.4097947E-02 1.2610607E+02 -2.1432196E+01 7.9234784E+02 1.0100551E+00 1.0000000E+03 -5.5426229E-02 1.2605644E+02 -2.1949785E+01 7.9203601E+02

NX Nastran Aeroelastic Analysis User’s Guide

Flutter Analysis Sample Problems

POINT = 10 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 3.9230 2.5490677E-01 5.0000000E+02 -8.4696122E-02 2.4974611E+02 -6.6452629E+01 1.5692012E+03 3.2690 3.0590245E-01 6.0000000E+02 -8.3397388E-02 2.4973441E+02 -6.5430573E+01 1.5691276E+03 2.8016 3.5693529E-01 7.0000000E+02 -8.2358606E-02 2.4970007E+02 -6.4606697E+01 1.5689119E+03 2.4509 4.0800592E-01 8.0000000E+02 -8.1597455E-02 2.4965118E+02 -6.3997078E+01 1.5686047E+03 2.1781 4.5911425E-01 9.0000000E+02 -8.1077963E-02 2.4959265E+02 -6.3574726E+01 1.5682369E+03 1.9598 5.1026040E-01 1.0000000E+03 -8.0758311E-02 2.4952739E+02 -6.3307526E+01 1.5678269E+03 POINT = 11 MACH NUMBER = 0.8000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 25.5828 3.9088804E-02 5.0000000E+02 4.3019064E-02 1.6286499E+03 2.2010939E+02 1.0233109E+04 21.3129 4.6920050E-02 6.0000000E+02 4.0013641E-02 1.6281818E+03 2.0467314E+02 1.0230168E+04 18.2631 5.4755293E-02 7.0000000E+02 3.7123404E-02 1.6277289E+03 1.8983650E+02 1.0227322E+04 15.9759 6.2594332E-02 8.0000000E+02 3.4157462E-02 1.6272905E+03 1.7462265E+02 1.0224568E+04 14.1971 7.0436925E-02 9.0000000E+02 3.1239031E-02 1.6268677E+03 1.5966130E+02 1.0221911E+04 12.7742 7.8282855E-02 1.0000000E+03 2.8289590E-02 1.6264602E+03 1.4455063E+02 1.0219352E+04 COMPLEX EIGENVALUE = -4.414635E+02, 0.000000E+00 C O M P L E X E I G E N V E C T O R NO. 1 (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 R2 1 G -5.622806E-26 0.0 5.553750E-04 0.0 2.939524E-05 0.0 0.0 0.0 0.0 0.0 2 G -7.804397E-26 0.0 1.599101E-04 0.0 2.030263E-05 0.0 0.0 0.0 0.0 0.0 3 G 8.587670E-23 0.0 -3.057504E-05 0.0 6.584304E-06 0.0 0.0 0.0 0.0 0.0 4 G 1.774474E-21 0.0 -1.539706E-04 0.0 1.613624E-05 0.0 0.0 0.0 0.0 0.0 5 G 3.197311E-21 0.0 -6.649786E-04 0.0 5.575138E-05 0.0 0.0 0.0 0.0 0.0 6 G 2.814081E-21 0.0 -1.805823E-03 0.0 9.122852E-05 0.0 0.0 0.0 0.0 0.0 7 G 1.467198E-20 0.0 -3.157421E-03 0.0 7.782308E-05 0.0 0.0 0.0 0.0 0.0 8 G 1.613662E-20 0.0 -3.870139E-03 0.0 1.034301E-05 0.0 0.0 0.0 0.0 0.0 9 G 1.902839E-20 0.0 -3.429607E-03 0.0 -6.475838E-05 0.0 0.0 0.0 0.0 0.0 10 G 1.888001E-20 0.0 -2.148297E-03 0.0 -9.572718E-05 0.0 0.0 0.0 0.0 0.0 11 G 1.895301E-20 0.0 -2.846956E-03 0.0 -8.800939E-05 0.0 0.0 0.0 0.0 0.0 12 G 1.895301E-20 0.0 -2.846956E-03 0.0 -8.800939E-05 0.0 0.0 0.0 0.0 0.0 21 G -8.290486E-20 0.0 1.151502E+00 1.092424E-07 7.695661E-02 0.0 0.0 0.0 0.0 0.0 22 G -1.294071E-19 -9.139968E-20 1.151503E+00 7.035015E-08 7.695664E-02 0.0 0.0 0.0 0.0 0.0 23 G 9.858746E-20 -1.053158E-19 1.151504E+00 1.014162E-07 7.695661E-02 0.0 0.0 0.0 0.0 0.0 24 G 1.895301E-20 0.0 -2.846956E-03 0.0 7.695661E-02 0.0 0.0 0.0 0.0 0.0 25 G 5.835389E-20 -7.615703E-20 -2.846302E-03 1.200169E-07 7.695661E-02 0.0 0.0 0.0 0.0 0.0 26 G 2.750808E-20 -8.930845E-20 -2.845197E-03 9.818396E-08 7.695661E-02 0.0 0.0 0.0 0.0 0.0 27 G 2.581491E-20 0.0 -1.157196E+00 1.092369E-07 7.695661E-02 0.0 0.0 0.0 0.0 0.0 28 G -1.098622E-19 -8.260402E-21 -1.157195E+00 7.034458E-08 7.695657E-02 0.0 0.0 0.0 0.0 0.0 29 G 1.397118E-19 -8.312845E-20 -1.157194E+00 1.014106E-07 7.695660E-02 0.0 0.0 0.0 0.0 0.0 45 G 0.0 0.0 -3.351959E-04 0.0 3.312885E-05 0.0 0.0 0.0 0.0 0.0 49 S 7.704461E-02 0.0 COMPLEX EIGENVALUE = -4.701534E+01, 0.000000E+00 C O M P L E X E I G E N V E C T O R NO. 2 (REAL/IMAGINARY) POINT ID. TYPE T1 T2 T3 R1 R2 1 G 7.697230E-27 0.0 -2.437863E-02 0.0 -9.502235E-04 0.0 0.0 0.0 0.0 0.0 2 G 6.255967E-27 0.0 -1.012796E-02 0.0 -9.496883E-04 0.0 0.0 0.0 0.0 0.0 3 G 8.878045E-23 0.0 4.118078E-03 0.0 -9.503659E-04 0.0 0.0 0.0 0.0 0.0 4 G 1.796950E-21 0.0 1.840974E-02 0.0 -9.563883E-04 0.0 0.0 0.0 0.0 0.0 5 G 3.186261E-21 0.0 3.283658E-02 0.0 -9.677802E-04 0.0 0.0 0.0 0.0 0.0 6 G 2.764943E-21 0.0 4.743726E-02 0.0 -9.779941E-04 0.0 0.0 0.0 0.0 0.0 7 G 1.488008E-20 0.0 6.213151E-02 0.0 -9.792305E-04 0.0 0.0 0.0 0.0 0.0 8 G 1.620515E-20 0.0 7.676443E-02 0.0 -9.705185E-04 0.0 0.0 0.0 0.0 0.0 9 G 1.993779E-20 0.0 9.123545E-02 0.0 -9.594839E-04 0.0 0.0 0.0 0.0 0.0 10 G 1.975076E-20 0.0 1.055806E-01 0.0 -9.547768E-04 0.0 0.0 0.0 0.0 0.0 11 G 1.984591E-20 0.0 9.841685E-02 0.0 -9.559511E-04

NX Nastran Aeroelastic Analysis User’s Guide

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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Chapter 7

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0.0 0.0 0.0 0.0 1.984591E-20 0.0 9.841685E-02 0.0 0.0 0.0 0.0 0.0 -1.810074E-19 0.0 7.926920E-01 -1.151756E-08 0.0 0.0 0.0 0.0 5.418016E-20 -4.200420E-19 7.926919E-01 -7.418621E-09 0.0 0.0 0.0 0.0 -9.360635E-20 -6.292123E-19 7.926918E-01 -1.069350E-08 0.0 0.0 0.0 0.0 1.984591E-20 0.0 9.841685E-02 0.0 0.0 0.0 0.0 0.0 1.004708E-19 -6.511857E-20 9.841678E-02 -1.265238E-08 0.0 0.0 0.0 0.0 2.804672E-20 -7.622544E-20 9.841666E-02 -1.035237E-08 0.0 0.0 0.0 0.0 -1.311941E-19 0.0 -5.958583E-01 -1.151652E-08 0.0 0.0 0.0 0.0 1.805717E-20 2.135124E-19 -5.958585E-01 -7.417567E-09 0.0 0.0 0.0 0.0 8.438422E-20 2.614343E-19 -5.958585E-01 -1.069243E-08 0.0 0.0 0.0 0.0 0.0 0.0 2.560180E-02 0.0 0.0 0.0 0.0 0.0 4.724097E-02 0.0 -5.875274E+00, 6.040307E+00 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY) T1 T2 T3 R1 -1.532115E-29 0.0 -5.033563E-02 0.0 -1.924545E-28 0.0 -4.450270E-02 0.0 -2.321368E-27 0.0 -1.891541E-02 0.0 8.086348E-29 0.0 -3.980703E-02 0.0 7.678913E-23 0.0 1.250183E-02 0.0 3.018241E-23 0.0 -3.511261E-02 0.0 1.565272E-21 0.0 4.391262E-02 0.0 -3.682051E-23 0.0 -3.042408E-02 0.0 2.711056E-21 0.0 7.531462E-02 0.0 -1.212795E-22 0.0 -2.574827E-02 0.0 2.293700E-21 0.0 1.067080E-01 0.0 -1.038983E-22 0.0 -2.108882E-02 0.0 1.298636E-20 0.0 1.380959E-01 0.0 -5.927381E-22 0.0 -1.644207E-02 0.0 1.394255E-20 0.0 1.694821E-01 0.0 -6.210600E-22 0.0 -1.179908E-02 0.0 1.826043E-20 0.0 2.008695E-01 0.0 -7.725466E-22 0.0 -7.152530E-03 0.0 1.804638E-20 0.0 2.322583E-01 0.0 -7.666137E-22 0.0 -2.501670E-03 0.0 1.815892E-20 0.0 2.165638E-01 0.0 -7.695926E-22 0.0 -4.827428E-03 0.0 1.815892E-20 0.0 2.165638E-01 0.0 -7.695926E-22 0.0 -4.827428E-03 0.0 -2.960137E-19 0.0 2.127821E-01 -1.309042E-10 5.393404E-21 0.0 -3.832612E-02 -5.705644E-10 3.100582E-19 -8.324270E-19 2.127821E-01 -8.414084E-11 7.112336E-22 1.088462E-20 -3.832613E-02 -3.672180E-10 -3.562011E-19 -1.292242E-18 2.127821E-01 -1.214148E-10 3.434035E-22 1.640910E-20 -3.832613E-02 -5.295384E-10 1.815892E-20 0.0 2.165638E-01 0.0 -7.695926E-22 0.0 -4.827428E-03 0.0 1.467939E-19 -3.988072E-20 2.165638E-01 -1.439346E-10 -3.668352E-21 1.775314E-21 -4.827431E-03 -6.270001E-10 2.468367E-20 -4.662222E-20 2.165638E-01 -1.175776E-10 -9.327292E-22 1.190369E-21 -4.827437E-03 -5.127050E-10 -3.378388E-19 0.0 2.203455E-01 -1.309458E-10 3.149701E-21 0.0 2.867127E-02 -5.706007E-10 1.992088E-19 5.016739E-19 2.203455E-01 -8.418290E-11 2.698738E-21 -5.275698E-21 2.867126E-02 -3.672549E-10 -7.272513E-21 7.193553E-19 2.203455E-01 -1.214573E-10 -4.814626E-21 -4.785630E-21 2.867126E-02 -5.295756E-10 0.0 0.0 5.961477E-02 0.0 0.0 0.0 -2.808424E-02 0.0 1.840479E-03 -1.923174E-03 -2.660025E+02, 2.693527E+02 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY) T1 T2 T3 R1 2.818845E-26 0.0 1.075129E-03 0.0 6.284259E-26 0.0 -4.932519E-04 0.0 3.228420E-26 0.0 2.858225E-04 0.0 8.214883E-26 0.0 -1.729764E-04 0.0 8.602143E-23 0.0 -5.777448E-04 0.0 -1.694287E-25 0.0 1.004960E-05 0.0 1.774438E-21 0.0 -1.422288E-03 0.0 1.545034E-25 0.0 2.065171E-04 0.0 3.200527E-21 0.0 -2.013585E-03 0.0 2.136683E-24 0.0 7.890490E-04 0.0 2.824258E-21 0.0 -2.244639E-03 0.0 6.538300E-24 0.0 1.859436E-03 0.0

NX Nastran Aeroelastic Analysis User’s Guide

0.0 -9.559511E-04 0.0 4.628501E-02 0.0 4.628501E-02 0.0 4.628501E-02 0.0 4.628501E-02 0.0 4.628501E-02 0.0 4.628501E-02 0.0 4.628501E-02 0.0 4.628501E-02 0.0 4.628501E-02 0.0 -9.616409E-04 0.0

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10 R2 -2.094713E-03 -3.130503E-04 -2.094619E-03 -3.130329E-04 -2.094303E-03 -3.128359E-04 -2.093773E-03 -3.122136E-04 -2.093161E-03 -3.111781E-04 -2.092665E-03 -3.101271E-04 -2.092427E-03 -3.095542E-04 -2.092438E-03 -3.096032E-04 -2.092548E-03 -3.099410E-04 -2.092605E-03 -3.101157E-04 -2.092590E-03 -3.100719E-04 -2.092590E-03 -3.100719E-04 -2.521118E-04 -2.233246E-03 -2.521118E-04 -2.233247E-03 -2.521118E-04 -2.233246E-03 -2.521118E-04 -2.233246E-03 -2.521118E-04 -2.233246E-03 -2.521118E-04 -2.233246E-03 -2.521118E-04 -2.233246E-03 -2.521117E-04 -2.233246E-03 -2.521118E-04 -2.233246E-03 -2.093466E-03 -3.117333E-04

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

11 R2 5.142017E-05 -2.352921E-05 5.502103E-05 -1.699670E-05 5.907048E-05 -9.144279E-06 5.031910E-05 -2.232489E-05 2.682101E-05 -5.735606E-05 6.866813E-06 -7.883458E-05

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

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1.469809E-20 0.0 -2.350602E-03 0.0 2.334632E-23 0.0 2.884958E-03 0.0 1.615941E-20 0.0 -2.779015E-03 0.0 1.762085E-23 0.0 2.993036E-03 0.0 1.901493E-20 0.0 -3.796912E-03 0.0 3.560084E-24 0.0 1.738618E-03 0.0 1.887053E-20 0.0 -5.244609E-03 0.0 5.837512E-24 0.0 -4.545770E-04 0.0 1.894110E-20 0.0 -4.491115E-03 0.0 4.430526E-24 0.0 7.057441E-04 0.0 1.894110E-20 0.0 -4.491115E-03 0.0 4.430526E-24 0.0 7.057441E-04 0.0 -6.899983E-20 0.0 1.154456E+00 -5.650440E-08 8.320644E-21 0.0 -7.100240E-03 -1.619360E-07 -1.552224E-19 -4.448372E-20 1.154456E+00 -3.638887E-08 -1.545112E-20 2.812469E-20 -7.101622E-03 -1.042826E-07 1.256605E-19 -3.073078E-20 1.154456E+00 -5.245712E-08 1.621189E-20 4.467221E-20 -7.102918E-03 -1.503340E-07 1.894110E-20 0.0 -4.491115E-03 0.0 4.430526E-24 0.0 7.057441E-04 0.0 5.221789E-20 -7.767380E-20 -4.491453E-03 -6.207656E-08 -3.682358E-21 -9.162501E-22 7.047749E-04 -1.779085E-07 2.752273E-20 -9.087362E-20 -4.492024E-03 -5.078503E-08 1.602703E-23 -9.082022E-22 7.031364E-04 -1.455429E-07 4.785045E-20 0.0 -1.163439E+00 -5.650123E-08 1.318674E-20 0.0 8.511524E-03 -1.619282E-07 -1.279969E-19 -3.971037E-20 -1.163439E+00 -3.638566E-08 -1.089097E-20 -1.884383E-20 8.510142E-03 -1.042748E-07 1.476247E-19 -1.321660E-19 -1.163440E+00 -5.245388E-08 4.773230E-21 -2.942496E-20 8.508846E-03 -1.503260E-07 0.0 0.0 -1.761995E-03 0.0 0.0 0.0 4.320996E-04 0.0 7.716534E-02 -6.694534E-04 -5.430004E+00, 2.823035E+02 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY) T1 T2 T3 R1 1.870345E-24 0.0 -3.213434E-01 0.0 -4.671143E-24 0.0 8.473271E-01 0.0 2.439922E-24 0.0 -1.326275E-01 0.0 -6.105403E-24 0.0 3.532941E-01 0.0 8.585457E-23 0.0 3.830048E-02 0.0 -7.311590E-24 0.0 -9.541247E-02 0.0 1.771356E-21 0.0 1.696936E-01 0.0 -9.301234E-24 0.0 -4.426189E-01 0.0 3.308824E-21 0.0 2.416040E-01 0.0 -2.712513E-22 0.0 -6.362043E-01 0.0 3.150580E-21 0.0 2.419822E-01 0.0 -8.064501E-22 0.0 -6.448789E-01 0.0 1.520942E-20 0.0 1.699723E-01 0.0 -1.382427E-21 0.0 -4.675196E-01 0.0 1.660866E-20 0.0 3.636296E-02 0.0 -1.168522E-21 0.0 -1.337333E-01 0.0 1.638194E-20 0.0 -1.390269E-01 0.0 6.320051E-21 0.0 3.045416E-01 0.0 1.640249E-20 0.0 -3.333826E-01 0.0 5.918725E-21 0.0 7.895467E-01 0.0 1.637660E-20 0.0 -2.351052E-01 0.0 6.153534E-21 0.0 5.443553E-01 0.0 1.637660E-20 0.0 -2.351052E-01 0.0 6.153534E-21 0.0 5.443553E-01 0.0 3.878907E-19 0.0 1.034723E+00 -1.014327E-05 -1.113556E-18 0.0 4.719849E-01 2.573992E-05 -1.002861E-18 1.487832E-18 1.034636E+00 -6.530712E-06 2.065285E-18 -3.734592E-18 4.722046E-01 1.657252E-05 1.016803E-18 2.405414E-18 1.034555E+00 -9.415642E-06 -2.171645E-18 -5.936928E-18 4.724105E-01 2.389344E-05 1.637660E-20 0.0 -2.351052E-01 0.0 6.153534E-21 0.0 5.443553E-01 0.0 -1.465893E-19 -1.303139E-19 -2.351659E-01 -1.114472E-05 4.848060E-19 1.286969E-19 5.445095E-01 2.828126E-05 2.550640E-20 -1.485272E-19 -2.352685E-01 -9.115832E-06 4.901439E-21 1.407833E-19 5.447699E-01 2.313264E-05 7.732950E-19 0.0 -1.504946E+00 -1.014317E-05 -1.767721E-18 0.0 6.167585E-01 2.573969E-05 -7.216190E-19 -1.066945E-18 -1.505033E+00 -6.530615E-06 1.446336E-18 2.503064E-18 6.169782E-01 1.657229E-05 4.044699E-19 -1.735715E-18 -1.505114E+00 -9.415544E-06 -6.257459E-19 3.908054E-18 6.171841E-01 2.389320E-05 0.0 0.0 2.140459E-01 0.0 0.0 0.0 -5.611298E-01 0.0 7.164872E-02 2.762995E-02 -2.053377E+01, 7.937534E+02 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY) T1 T2 T3 R1 1.557226E-23 0.0 7.371728E-01 0.0 1.755097E-24 0.0 8.454187E-02 0.0 2.034889E-23 0.0 -1.081655E-01 0.0

1.311525E-05 -4.648737E-05 4.754812E-05 3.806402E-05 8.595373E-05 1.234993E-04 1.017805E-04 1.575336E-04 9.783642E-05 1.490612E-04 9.783642E-05 1.490612E-04 7.726318E-02 -5.203916E-04 7.726315E-02 -5.204456E-04 7.726317E-02 -5.204011E-04 7.726318E-02 -5.203921E-04 7.726318E-02 -5.203921E-04 7.726318E-02 -5.203921E-04 7.726318E-02 -5.203927E-04 7.726319E-02 -5.203386E-04 7.726318E-02 -5.203831E-04 3.984460E-05 -3.833295E-05

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

12 R2 -1.276668E-02 3.340675E-02 -1.220983E-02 3.199311E-02 -1.032278E-02 2.716874E-02 -6.963280E-03 1.851454E-02 -2.484811E-03 6.931479E-03 2.444806E-03 -5.788450E-03 7.031926E-03 -1.751461E-02 1.055133E-02 -2.638222E-02 1.256257E-02 -3.136872E-02 1.315202E-02 -3.281040E-02 1.300692E-02 -3.245573E-02 1.300692E-02 -3.245573E-02 8.465567E-02 -4.825869E-03 8.465228E-02 -4.817276E-03 8.465508E-02 -4.824358E-03 8.465564E-02 -4.825788E-03 8.465564E-02 -4.825788E-03 8.465564E-02 -4.825788E-03 8.465561E-02 -4.825706E-03 8.465899E-02 -4.834298E-03 8.465620E-02 -4.827217E-03 -4.829020E-03 1.299704E-02

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

13 R2 5.957617E-02 6.881985E-03 4.991532E-02

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2.293953E-24 0.0 -1.306932E-02 0.0 1.029616E-22 0.0 -6.703025E-01 0.0 2.329235E-24 0.0 -7.779325E-02 0.0 1.792482E-21 0.0 -7.360933E-01 0.0 2.037106E-24 0.0 -8.513510E-02 0.0 3.936770E-21 0.0 -3.217615E-01 0.0 8.407100E-23 0.0 -3.737912E-02 0.0 5.083291E-21 0.0 2.970941E-01 0.0 2.579491E-22 0.0 3.339669E-02 0.0 1.854614E-20 0.0 7.283687E-01 0.0 4.267467E-22 0.0 8.221842E-02 0.0 2.019816E-20 0.0 6.832948E-01 0.0 4.547182E-22 0.0 7.643548E-02 0.0 1.152795E-20 0.0 1.321284E-01 0.0 -8.864207E-22 0.0 1.306858E-02 0.0 1.234611E-20 0.0 -7.124583E-01 0.0 -7.759008E-22 0.0 -8.353228E-02 0.0 1.184123E-20 0.0 -2.714428E-01 0.0 -8.419210E-22 0.0 -3.311513E-02 0.0 1.184123E-20 0.0 -2.714428E-01 0.0 -8.419210E-22 0.0 -3.311513E-02 0.0 2.883387E-18 0.0 9.329480E-01 -9.867944E-05 3.375858E-19 0.0 -1.726366E-02 -1.132793E-05 -5.791094E-18 9.996100E-18 9.321057E-01 -6.355390E-05 -6.446770E-19 1.148109E-18 -1.736035E-02 -7.295685E-06 6.002801E-18 1.593555E-17 9.313161E-01 -9.161426E-05 6.722199E-19 1.825749E-18 -1.745099E-02 -1.051688E-05 1.184123E-20 0.0 -2.714428E-01 0.0 -8.419210E-22 0.0 -3.311513E-02 0.0 -1.233495E-18 -4.640567E-19 -2.720334E-01 -1.084075E-04 -1.469619E-19 -4.429292E-20 -3.318292E-02 -1.244465E-05 3.356662E-20 -5.021118E-19 -2.730317E-01 -8.869314E-05 6.777912E-22 -4.722107E-20 -3.329753E-02 -1.018155E-05 4.867381E-18 0.0 -1.475958E+00 -9.867260E-05 5.513232E-19 0.0 -4.898085E-02 -1.132714E-05 -4.098088E-18 -6.877520E-18 -1.476800E+00 -6.354698E-05 -4.540927E-19 -7.820732E-19 -4.907754E-02 -7.294891E-06 1.934026E-18 -1.080533E-17 -1.477590E+00 -9.160728E-05 2.043801E-19 -1.220715E-18 -4.916817E-02 -1.051608E-05 0.0 0.0 -5.761115E-01 0.0 0.0 0.0 -6.663808E-02 0.0 2.314919E-02 -5.478018E-03 -6.645263E+01, 1.569201E+03 C O M P L E X E I G E N V E C T O R NO. (REAL/IMAGINARY) T1 T2 T3 R1 6.401050E-23 0.0 -6.276693E-01 0.0 -4.878966E-24 0.0 4.651555E-02 0.0 8.365917E-23 0.0 5.030121E-01 0.0 -6.376972E-24 0.0 -3.418593E-02 0.0 1.499636E-22 0.0 7.852564E-01 0.0 -6.489951E-24 0.0 -5.317898E-02 0.0 1.841860E-21 0.0 9.303456E-02 0.0 -6.909731E-24 0.0 -2.211586E-03 0.0 5.143534E-21 0.0 -7.345554E-01 0.0 -1.504227E-22 0.0 5.632180E-02 0.0 9.030332E-21 0.0 -7.512636E-01 0.0 -4.788753E-22 0.0 5.353622E-02 0.0 3.843673E-20 0.0 5.847566E-02 0.0 -1.798759E-21 0.0 -9.385129E-03 0.0 3.423218E-20 0.0 7.636663E-01 0.0 -1.377613E-21 0.0 -6.132064E-02 0.0 3.100526E-20 0.0 5.090676E-01 0.0 -8.391481E-22 0.0 -3.839652E-02 0.0 3.290901E-20 0.0 -6.045071E-01 0.0 -9.985415E-22 0.0 5.046965E-02 0.0 3.168738E-20 0.0 1.471206E-02 0.0 -8.981993E-22 0.0 1.260879E-03 0.0 3.168738E-20 0.0 1.471206E-02 0.0 -8.981993E-22 0.0 1.260879E-03 0.0 8.361051E-18 0.0 1.151624E+00 1.356225E-05 -6.491242E-19 0.0 3.332649E-03 -3.664619E-07 -1.548839E-17 2.827968E-17 1.151739E+00 8.742995E-06 1.182386E-18 -2.181730E-18 3.329518E-03 -2.366489E-07 1.627306E-17 4.497190E-17 1.151848E+00 1.259704E-05 -1.244800E-18 -3.466541E-18 3.326580E-03 -3.406660E-07 3.168738E-20 0.0 1.471206E-02 0.0 -8.981993E-22 0.0 1.260879E-03 0.0 -3.715567E-18 -8.624652E-19 1.479318E-02 1.489300E-05 2.899163E-19 6.120346E-20 1.258689E-03 -4.021145E-07 3.982245E-20 -8.513324E-19 1.493039E-02 1.219370E-05 -9.543344E-22 5.960965E-20 1.254982E-03 -3.296754E-07 1.311267E-17 0.0 -1.122183E+00 1.355879E-05 -1.007602E-18 0.0 -8.113220E-04 -3.662450E-07 -1.089529E-17 -1.880055E-17 -1.122067E+00 8.739498E-06 8.303002E-19 1.446313E-18 -8.144504E-04 -2.364298E-07 4.748255E-18 -2.937522E-17 -1.121958E+00 1.259351E-05 -3.553317E-19 2.254490E-18 -8.173860E-04 -3.404446E-07 0.0 0.0 -3.733928E-01 0.0 0.0 0.0 3.121659E-02 0.0 -1.218172E-03 6.274760E-03

NX Nastran Aeroelastic Analysis User’s Guide

5.758268E-03 2.225761E-02 2.551969E-03 -1.336020E-02 -1.552989E-03 -3.852503E-02 -4.422840E-03 -3.917717E-02 -4.462009E-03 -1.479234E-02 -1.648630E-03 2.110641E-02 2.448772E-03 4.961475E-02 5.683510E-03 5.962925E-02 6.815802E-03 5.714767E-02 6.535258E-03 5.714767E-02 6.535258E-03 8.029722E-02 1.057282E-03 8.026426E-02 1.053498E-03 8.029139E-02 1.056612E-03 8.029686E-02 1.057240E-03 8.029686E-02 1.057240E-03 8.029686E-02 1.057239E-03 8.029650E-02 1.057198E-03 8.032946E-02 1.060982E-03 8.030233E-02 1.057867E-03 -2.846188E-02 -3.281641E-03

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14 R2 -8.624890E-02 6.172903E-03 -5.363850E-02 3.794490E-03 1.821774E-02 -1.411957E-03 6.276304E-02 -4.541531E-03 3.469245E-02 -2.322076E-03 -3.267320E-02 2.680272E-03 -6.260585E-02 4.742374E-03 -1.988939E-02 1.321605E-03 5.204502E-02 -4.226515E-03 8.533797E-02 -6.773440E-03 7.701172E-02 -6.136628E-03 7.701172E-02 -6.136628E-03 7.579347E-02 1.381354E-04 7.579801E-02 1.380123E-04 7.579429E-02 1.381123E-04 7.579355E-02 1.381324E-04 7.579355E-02 1.381323E-04 7.579355E-02 1.381322E-04 7.579362E-02 1.381293E-04 7.578908E-02 1.382523E-04 7.579280E-02 1.381522E-04 5.839512E-02 -4.137437E-03

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

Listing 7-46. Output for Aeroservoelastic Stability Analysis of a Missile

7.14 Aerothermoelastic Stability of a Wing (Examples HA153A and HA145KR) The field of aerothermoelasticity considers the effects of thermally induced stresses on structural stiffness and their aeroelastic interaction. There are two mechanisms by which structural stiffness is reduced at high temperature: the first is the reduction of structural moduli of elasticity, and the second is the reduction in stiffness from thermally induced compressive stresses (beam-column effects). The former effect is neglected in this example. The latter effect is considered here by determining the stiffness matrix as a function of temperature using the NX Nastran SOLution 153, Static Structural and/or Steady State Heat Transfer Analysis with Options: Linear or Nonlinear Analysis. With the temperature-dependent stiffness matrix, restarts can be made in the various Aeroelastic Solution Sequences. SOL 145, Aerodynamic Flutter, is selected here, and the variation of flutter speed with temperature is determined. Restarts in SOL 144, Static Aeroelastic Response, and SOL 146, Dynamic Aeroelastic Response, solutions are also discussed. Interest in supersonic transport aircraft led to an extensive literature related to aerothermoelasticity in the late 1950s, some examples of which include Budiansky and Mayers (1956), Bisplinghoff (1956), Turner, Dill, Martin, and Melosh (1960), and a Proceedings of Symposium on Aerothermoelasticity (Anon., 1961). A recent assessment of analysis and testing applicable to the National Aero-Space Plane is presented by Friedman et al. (1988) and contains additional references. The example is a small wing that might be typical of a cantilevered wind tunnel model (see Figure 7-22). It has an unswept, rectangular planform with an internal structure that consists of three spars connected only by ribs at the root and at the tip. In order to emphasize thermoelastic effects, the closing ribs are assumed to be rigid, the main (center) spar is not heated, and the forward and aft spars are assumed to be heated uniformly to the same temperature. The spars are assumed to be weightless but support a number of concentrated weights. The following geometry and other model properties are assumed for the input to SOL 153. The rectangular lifting surface extends forward and aft of the structural model. A wing span of 100.0 in. and a wing chord of 40.0 in. are assumed. The wing spars are spaced 10.0 in apart and are located within the planform such that the wing leading edge is 5.0 in. forward of the front spar and the trailing edge is 15.0 in. behind the rear spar. The spars are thus located at 12.5%, 37.5%, and 62.5% of the wing chord. There are no intermediate ribs connecting the spars between the root and the tip. The three spars are assumed to be uniform across the span and to have the same section properties, each being relatively stiff in the chordwise direction but weak in torsion. The spar material is assumed to be aluminum with moduli E = 10.3 × 106 and G = 3.9 × 106 psi, and coefficient of thermal expansion α = 1.244 × 105 in/in/°F. The spar section properties are assumed to be: area A = 2.0 in2, Iy = 1.0 in4, Iz = 100.0 in4, Iyz = 0.0, and J = 0.1 in4.

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Figure 7-22. Structural and Aerodynamic Models for Aerothermoelastic Example The NX Nastran structural model consists of GRIDs, BEAMs (note that SOL 153 does not support the NX Nastran BAR element), RBARs, and CONM2s. The structural layout and the aerodynamic boxes are shown in Figure 7-22. Each spar is modeled by five equal length BEAM elements. The leading edge spar has GRIDs 100 through 105 and CBEAMs 101 through 105, the main spar has GRIDs 200 through 205 and CBEAMs 201 through 205, and the aft spar has GRIDs 300 through 305 and CBEAMs 301 through 305. Each spar element has the same properties listed on PBEAM 100 with material properties on MAT1 1. Although no variation of material properties with temperature is considered here, provision is made for variations with temperature by including the MATT1 entry with its associated TABLEMi entries using the same properties at both ends of the tabulated temperature range. Root GRIDs 100 and 200 are rigidly connected by rigid bar RBAR 100, and GRIDs 200 and 300 are connected by RBAR 200. Tip GRIDs 105 and 205 are connected by RBAR 105, and GRIDs 205 and 305 are connected by RBAR 205.

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Flutter Analysis Sample Problems

Constraints are an important consideration in this restart analysis since statically determinate constraints are desirable in SOL 153, but the different NX Nastran aeroelastic SOLs may require different constraints. The flutter analysis of this example assumes a cantilever condition, but a static aeroelastic analysis requires a SUPORT entry, so the specification of the cantilever condition must be completed in the restart flutter analysis. For these reasons, the initial constraints imposed on the structure are only as many as the aeroelastic analyses permit. For symmetric aeroelastic analyses these are DOFs T1, T2, R1, and R3 and are imposed in the permanent constraint field on GRID 200 (1246). The additional cantilever constraints (DOFs T3, and R2, i.e., 35) for the flutter analysis will be included in the Bulk Data for the restart in SOL 145 using the database from SOL 153. For the symmetric static aeroelastic analysis, GRID 200 would be SUPORTed in DOFs T3 and R2 in the restart in SOL 144 from the SOL 153 database. The PARAM,WTMASS is included in the Bulk Data for SOL 153 for convenience since it is needed in all of the aeroelastic solutions to convert input weight to mass units. The PARAM,LGDISP,1 is required in SOL 153 to account for large displacement effects in updating element coordinates and follower forces. The temperature data are input on TEMP and TEMPD entries. The TEMPD entry is used to specify all temperatures not specified on the TEMP entry and is also used here to specify the initial temperature. Four sets of TEMP entries are used to specify the increase in temperature on the forward and aft spars. TEMP 2 specifies a slight temperature increase of 0.001°F to simulate the cold condition. TEMPs 3, 4, and 5 specify temperatures of 50, 150, and 200°F. The nonlinear parameter entry NLPARM specifies data for the nonlinear iteration strategy: NINC is the number of increments to reach the final temperature; KMETHOD is the method for controlling the stiffness updates; KSTEP is the number of iterations between stiffness updates; CONV selects the convergence criteria; and INTOUT provides intermediate output. For this example, the number of increments NINC is chosen depending on the temperature change: one increment to reach 0.001°F, 4 to go from 0.001 to 50°F, 10 to go from 50 to 150°F, and 10 to go from 150 to 200°F. The other parameters are selected as KMETHOD = AUTO (the default), KSTEP = 1, CONV = PW (the default), and INTOUT = YES. At the end of the Bulk Data is the CORD2R 2 entry. This is the coordinate system for the spline that connects the aerodynamic and structural grids in the aeroelastic solutions. Normally this coordinate system would be contained in the Bulk Data for the aeroelastic solutions but is placed here to avoid the requirement for a DMAP Alter in the aeroelastic solutions that would skip the coordinate system check in the restart.

SOL 153 Case Control Command The Case Control Section for SOL 153 specifies the title and subtitle, requests printout of both annotated and sorted Bulk Data entries, and prescribes the initial temperature, before specifying the four subcases to increase the temperatures on the front and rear spars by 200°F. The Executive Control Section for SOL 153 specifies the problem ID NXN, HA153A, five minutes of CPU time, and SOL 153 for the Nonlinear Statics and Heat Transfer solution.

SOL 153 Output The input data are shown in Listing 7-47, the sorted Bulk Data entries are in Listing 7-48, and selected output data follow in Listing 7-49. The results of interest for the restart are discussed as follows. Because the structure in SOL 153 is not supported in a statically determinate manner, the stiffness matrix is not well conditioned. This is indicated at the beginning of the output by a warning message regarding excessive matrix factor diagonal ratios. This is not a problem in this example because the restarts in the Aeroelastic SOLs will provide sufficient additional supports.

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The remaining output shows the results from the nonlinear iteration module. Due to the purely axial loading that results from the temperature increase in this example, the structure behaves as geometrically linear in every load step. Therefore, the solution converges in one iteration step for every temperature increment. Of concern for the restart is, first, the fact of convergence and, then, the LOOPID number for the stiffness matrix that is saved for the restart. Twenty-five loops were executed and 25 stiffness matrices are saved on the database corresponding to various temperatures from 0.001 to 200°F. The flutter analysis is to be demonstrated at 100°F, and this temperature corresponds to Load Factor 0.500 in Subcase 3 with a LOOPID of 10. To obtain the stiffness matrix corresponding to the temperature of 100°F in the restart it is necessary to add the PARAM,NMLOOP and give it the value of the LOOPID for the temperature of interest, in this case, 10.

SOL 145 Bulk Data Entry The Bulk Data entries for the restart of Example HA145KR (R for Restart) in SOL 145, Aerodynamic Flutter Analysis, are considered next, and are shown in Listing 7-50. PARAM,GRDPNT,100 requests the weight and balance analysis relative to the root of the front spar at GRID 100. SPC1 imposes the additional constraints to achieve the cantilever boundary condition at the root of the main spar, GRID 200: the DOFs T3 and R2 are also constrained. The aerodynamic data begin with the AERO entry that specifies the reference chord as 20.0 in., sea-level density, and symmetric motion. The CAERO1 1001 entry specifies the unswept rectangular aerodynamic model for the subsonic Doublet-Lattice method with eight equal spanwise divisions and four equal chordwise divisions [see Figure 7-22(b)]; the continuation locates the root and tip leading edges at 5.0 in. forward of the forward spar, the 40.0 in. root and tip chords, and the 100.0 in. span. The PAERO1 1000 entry is required even though there is no fuselage. Linear SPLINE2 1001 connects aerodynamic boxes 1001 through 1032 to the normal structural translations (T3) only along the leading and trailing edge spars via the SET1 1100 GRIDs; a rigid chord is thereby effectively imposed even though there is no chordwise stiffening (ribs) except at the root and tip. The spline axis is aligned with the main spar using the coordinate system CORD2R 1. The MKAERO1 entry assumes incompressible flow, m = 0.0, and a range of reduced frequencies, k from 0.001 to 1.0. The flutter analysis is the final consideration. The vibration analysis for the modal flutter analysis is prescribed on the EIGR entry as the MGIV method, requesting four modes and normalizing the eigenvectors on their maximum components. The FLUTTER entry prescribes the PK-method, FLFACT entries for density, Mach number, and velocities, the output for four flutter modes, and a relaxed convergence criterion, ε = 0.01, for the PK-iteration. PARAM,LMODES,4 specifies all four vibration modes to be used in the flutter analysis, and PARAM,VREF,12.0 outputs velocity in ft/s in the Flutter Summary Tables. The Case Contol Section for the restart must include some information from the initial run. After the title, subtitle, and echo commands, the initial temperature must be given again. The PARAM,NMLOOP was mentioned above. It may be included either in the Bulk Data or in the Case Control. It is shown here in Case Control, and its value of 10 corresponds to the stiffness for 100°F. The additional constraints are called via the SPC command. The METHOD command invokes the EIGR entry, and the SVEC command requests vibration mode shapes. The FLUTTER command requests the PK-flutter analysis. The remaining data specify the plotting formats for the V-g and V-f curves. Finally, the BEGIN BULK command concludes the section. The Executive Control Section begins with the RESTART statement that specifies the database version to be used. Next are the problem identification, estimated CPU time, and SOL 145 for the flutter analysis. CEND is the last statement in the section.

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The input to the restart is shown in Listing 7-50; the sorted Bulk Data are in Listing 7-51, and the output from the flutter analysis is shown in Listing 7-52. The plotted V-g and V-f curves are shown in Figure 7-23. The results of special interest are the following. The weight and balance analysis shows a total wing weight of 250 lbs, with a center of gravity 60.0 in. outboard from the root and 14.0 in. aft of the front spar. The chordwise position of the center of gravity can also be expressed as 47.5% of the wing chord. The vibration analysis leads to the first four frequencies of 2.815, 5.649, 17.572, and 27.162 Hz. An examination of the eigenvectors indicates that the first mode is bending and the second mode is torsion. The third and fourth modes are difficult to characterize, but the spars appear to be vibrating somewhat independently due to the lack of chordwise stiffening. The flutter analysis predicts a second mode instability in Point 2 of the Flutter Summary (a typical bending/torsion flutter because of the close bending and torsion frequencies and aft center of gravity) at a velocity of 155 ft/s and frequency of 3.96 Hz. An instability also appears in Point 1 with a zero frequency at a velocity of 270 ft/s; this is the divergence speed. Both instabilities are seen in Figure 7-23. The above is a demonstration of the restart for flutter analysis with the front and rear spar temperatures raised 100°F. The vibration and flutter characteristics can be determined for other temperature increases by restarting with other values of PARAM,NMLOOP. The following values, from Listing 7-49, of LOOPID = NMLOOP correspond to the temperature increases noted: NMLOOP = 1 for 0.001°F, NMLOOP = 5 for 50°F, NMLOOP = 20 for 150°F, and NMLOOP = 25 for 200°F. The natural frequencies and flutter speeds have been studied up to 150°F and are shown in Figure 7-24 and Figure 7-25. The dramatic effect of temperature is apparent in the rapid decrease in frequencies and flutter speeds with increasing temperatures. Another study (not shown) of vibration frequencies above 150°F suggests thermally induced buckling around 180°F; this characteristic deserves further investigation.

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Figure 7-23. V-g and V-f Curves Illustrating Flutter and Divergence Characteristics

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Figure 7-24. Thermoelastic Effect on Cantilever Vibration Frequencies

Figure 7-25. Thermoelastic Effect on Flutter Speeds Restarts in the other aeroelastic SOLs, i.e., 144 and 146, can be made in the same way as the SOL 145 restart. Case Control must include the initial temperature TEMP(INIT) set number,

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and PARAM,NMLOOP must give the LOOPID for the temperature of interest. The Bulk Data must include any additional constraints for the prescribed boundary conditions, and SOL 144 must include the SUPORT entry. Any problems with the conditioning of the stiffness matrix from SOL 153 when restarting in SOL 144 can be overcome by including PARAM,BAILOUT,-1 in the Bulk Data. If there is any mass added to the system in the restart Bulk Data, e.g., an unrestrained fuselage mass, a DMAP Alter is required to skip the mass check in the restart. This example has used SOL 153 assuming the temperatures are known. If transient temperatures are obtained using SOL 159, the TEMP entries required by SOL 153 can be obtained in a “PCH” file by including the THERM(PRINT, PUNCH), OTIME, and SET commands appropriately in the SOL 159 Case Control Section and the PARAM,CURVPLOT,1 entry in the Bulk Data. N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA153A TIME 5 $ CPU TIME IN MINUTES SOL 153 $ NLSCSH, NONLINEAR STATICS AND HEAT TRANSFER CEND EXAMPLE HA153A: THREE BAR HEATED WING GEOMETRIC NONLINEAR ANALYSIS

PAGE C A S E

CARD COUNT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

C O N T R O L

D E C K

E C H O

TITLE = EXAMPLE HA153A: THREE BAR HEATED WING SUBTI = GEOMETRIC NONLINEAR STATICS AND HEAT TRANSFER ANALYSIS ECHO = BOTH TEMP(INIT) = 1 $ INITIAL TEMPERATURES SUBCASE 1 LABEL = TEMPERATURE LOAD TEMP(LOAD) = 2 $ TEMPERATURE INCREASE TO 0.001 DEG F. NLPARM = 10 SUBCASE 2 LABEL = TEMPERATURE LOAD TEMP(LOAD) = 3 $ TEMPERATURE INCREASE TO 50 DEG F. NLPARM = 20 SUBCASE 3 LABEL = TEMPERATURE LOAD TEMP(LOAD) = 4 $ TEMPERATURE INCREASE TO 150 DEG F. NLPARM = 30 SUBCASE 4 LABEL = TEMPERATURE LOAD TEMP(LOAD) = 5 $ TEMPERATURE INCREASE TO 200 DEG F. NLPARM = 40 BEGIN BULK

EXAMPLE HA153A: THREE BAR HEATED WING GEOMETRIC NONLINEAR ANALYSIS

PAGE I N P U T

. 1 $ $ $ $ $ $ GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID GRID $ $ $ $

7-186

3

..

2

..

3

..

B U L K 4

..

D A T A 5

..

6

4

D E C K ..

7

E C H O ..

8

..

9

..

THE GRID ENTRY DEFINES THE LOCATION OF A GEOMETRIC GRID POINT, THE DIRECTIONS OF ITS DISPLACEMENT, AND ITS PERMANENT SINGLE-POINT CONSTRAINTS. ID 100 101 102 103 104 105 200 201 202 203 204 205 300 301 302 303 304 305

CP

X1 0. 0. 0. 0. 0. 0. 10. 10. 10. 10. 10. 10. 20. 20. 20. 20. 20. 20.

X2 0. 20. 40. 60. 80. 100. 0. 20. 40. 60. 80. 100. 0. 20. 40. 60. 80. 100.

X3 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

CD

* * STRUCTURAL STIFFNESS PROPERTIES * *

NX Nastran Aeroelastic Analysis User’s Guide

PS

SEID

1246

$ $ $

10

.

Flutter Analysis Sample Problems

$ $ $ CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM CBEAM $ $ THE $ $ PBEAM $+PB1 +PB100 $ $ $ $ $ RBAR RBAR RBAR RBAR $ $ THE $ $ $ MAT1 $ $

THE CBEAM ENTRY DEFINES A BEAM ELEMENT. $ EID 101 102 103 104 105 201 202 203 204 205 301 302 303 304 305

PID 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

GA 100 101 102 103 104 200 201 202 203 204 300 301 302 303 304

GB 101 102 103 104 105 201 202 203 204 205 301 302 303 304 305

X1 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

X2 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

X3 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. $

PBEAM ENTRY DEFINES THE PROPERTIES OF A BEAM ELEMENT. PID 100 SO NO

MID 1 X/XB 1.0

A(A) 2.0 A

I1(A) 1.0 I1

I2(A) 100. I2

I12(A) 0.0 I12

J(A) 0.1 J

NSM(A)

$ +PB1 +PB100

NSM $

THE RBAR ENTRY DEFINES A RIGID BAR WITH SIX DEGREES OF FREEDOM AT EACH END. $ EID 100 105 200 205

GA 200 105 200 205

GB 100 205 300 305

CNA 123456 123456 123456 123456

CNB

CMA

CMB

$ MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES FOR LINEAR, TEMPERATURE-INDEPENDENT, ISOTROPIC MATERIALS. MID 1

E 10.3+6

G 3.9+6

NU

RHO

A TREF 1.244-5 0.0

$ $ $ GE $

THE MATT1 ENTRY SPECIFIES TEMPERATURE-DEPENDENT PROPERTIES $ ON MAT1 ENTRY FIELDS VIA TABLEMI ENTRIES. $ $ MID T(E) T(G) T(NU) T(RHO) T(A) T(GE) MATT1 1 1 2 3 $ TABLEM1 1 +TABM1 0.0 10.3+6 200.0 10.3+6 ENDT TABLEM1 2 +TABM2 0.0 3.9+6 200.0 3.9+6 ENDT TABLEM1 3 +TABM3 0.0 1.244-5 200.0 1.244-5 ENDT $ $ * * MASS AND INERTIA PROPERTIES * * $ $ THE CONM2 ENTRY DEFINES A CONCENTRATED MASS AT A GRID POINT. $ $ EID G CID M X1 X2 X3 CONM2 101 101 0 5.0 CONM2 102 102 0 5.0 CONM2 103 103 0 5.0 CONM2 104 104 0 5.0 CONM2 105 105 0 5.0 CONM2 201 201 0 20.0 CONM2 202 202 0 20.0 CONM2 203 203 0 20.0 CONM2 204 204 0 20.0 CONM2 205 205 0 20.0 CONM2 301 301 0 25.0 CONM2 302 302 0 25.0 CONM2 303 303 0 25.0 CONM2 304 304 0 25.0 CONM2 305 305 0 25.0 $ $ THE PARAM,WTMASS,GINV CAUSES ALL THE STRUCTURAL MASSES AND $ MASS DENSITIES TO BE MULTIPLIED BY GINV (I.E., BY ONE OVER $ THE ACCELERATION OF GRAVITY). THE DYNAMIC PRESSURE SUPPLIED $ FOR AERODYNAMIC FORCE CALCULATIONS WILL NOT BE MULTIPLIED $ BY GINV. $ PARAM WTMASS .0025907 $ $ $ * * * REQUEST GEOMETRIC NONLINEAR EFFECTS * * * $ $ IF LGDISP = 1, ALL THE NONLINEAR ELEMENT TYPES THAT HAVE A $ LARGE DISPLACEMENT CAPABILITY WILL BE ASSUMED TO HAVE LARGE $ DISPLACEMENT EFFECTS, I.E., UPDATED ELEMENT COORDINATES $ AND FOLLOWER FORCES. IF LGDISP = -1 (DEFAULT), THEN NO LARGE $ DISPLACEMENT EFFECTS WILL BE CONSIDERED.

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+TABM1 +TABM2 +TABM3 $ $ $

$ $ $ $ $ $ $

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$ PARAM $ $ $ $ $ $ $ $ $ $ TEMPD TEMPD TEMP TEMP TEMP TEMP TEMPD TEMP TEMP TEMP TEMP TEMPD TEMP TEMP TEMP TEMP TEMPD TEMP TEMP TEMP TEMP

LGDISP

1 *

*

*

TEMPERATURE DATA

*

*

*

THE TEMP ENTRY DEFINES THE TEMPERATURE AT GRID POINTS FOR DETERMINATION OF THERMAL LOADING, TEMPERATURE-DEPENDENT MATERIAL PROPERTIES, OR STRESS RECOVERY. THE TEMPD ENTRY DEFINES A TEMPERATURE VALUE FOR ALL GRID POINTS OF THE STRUCTURAL MODEL WHICH HAVE NOT BEEN GIVEN A TEMPERATURE ON A TEMP ENTRY. 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5

0. 0. 100 103 300 303 0. 100 103 300 303 0. 100 103 300 303 0. 100 103 300 303

0.001 0.001 0.001 0.001

101 104 301 304

0.001 0.001 0.001 0.001

102 105 302 305

0.001 0.001 0.001 0.001

50. 50. 50. 50.

101 104 301 304

50. 50. 50. 50.

102 105 302 305

50. 50. 50. 50.

150. 150. 150. 150.

101 104 301 304

150. 150. 150. 150.

102 105 302 305

150. 150. 150. 150.

200. 200. 200. 200.

101 104 301 304

200. 200. 200. 200.

102 105 302 305

200. 200. 200. 200.

$ $ * * * NONLINEAR SOLUTION PARAMETERS * * * $ $ THE NLPARM ENTRY DEFINES A SET OF PARAMETERS FOR NONLINEAR $ STATIC ANALYSIS ITERATION STRATEGY. $ $ ID NINC DT KMETHOD KSTEP MAXITER CONV INTOUT NLPARM 10 1 AUTO 1 PW YES NLPARM 20 4 AUTO 1 PW YES NLPARM 30 10 AUTO 1 PW YES NLPARM 40 10 AUTO 1 PW YES $ $ THE CORD2R ENTRY DEFINES A RECTANGULAR COORDINATE SYSTEM USING $ THE COORDINATES OF THREE POINTS. $ $ CID RID A1 A2 A3 B1 B2 B3 CORD2R 1 0 20. 0. 0. 20. 0. 10. $+CD C1 C2 C3 +CRD 30. 0. 0. $ ENDDATA INPUT BULK DATA CARD COUNT = 173

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+CD +CRD

Flutter Analysis Sample Problems

Listing 7-47. Input Data for Example HA153A EXAMPLE HA153A: THREE BAR HEATED WING GEOMETRIC NONLINEAR STATICS AND HEAT TRANSFER ANALYSIS

PAGE

S O R T E D CARD COUNT 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051-

MAT1 52535455565758596061626364656667686970717273747576777879808182-

. 1 .. 2 CBEAM 101 CBEAM 102 CBEAM 103 CBEAM 104 CBEAM 105 CBEAM 201 CBEAM 202 CBEAM 203 CBEAM 204 CBEAM 205 CBEAM 301 CBEAM 302 CBEAM 303 CBEAM 304 CBEAM 305 CONM2 101 CONM2 102 CONM2 103 CONM2 104 CONM2 105 CONM2 201 CONM2 202 CONM2 203 CONM2 204 CONM2 205 CONM2 301 CONM2 302 CONM2 303 CONM2 304 CONM2 305 CORD2R 1 +CRD 30. GRID 100 GRID 101 GRID 102 GRID 103 GRID 104 GRID 105 GRID 200 GRID 201 GRID 202 GRID 203 GRID 204 GRID 205 GRID 300 GRID 301 GRID 302 GRID 303 GRID 304 GRID 305 1

10.3+6 3.9+6 MATT1 1 NLPARM 10 NLPARM 20 NLPARM 30 NLPARM 40 PARAM LGDISP PARAM WTMASS PBEAM 100 +PB100 NO RBAR 100 RBAR 105 RBAR 200 RBAR 205 TABLEM1 1 +TABM1 0.0 TABLEM1 2 +TABM2 0.0 TABLEM1 3 +TABM3 0.0 TEMP 2 TEMP 2 TEMP 2 TEMP 2 TEMP 3 TEMP 3 TEMP 3 TEMP 3 TEMP 4 TEMP 4 TEMP 4 TEMP 4

.. 3 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 101 102 103 104 105 201 202 203 204 205 301 302 303 304 305 0 0.

.. 4 100 101 102 103 104 200 201 202 203 204 300 301 302 303 304 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20. 0. 0. 0. 0. 0. 0. 0. 10. 10. 10. 10. 10. 10. 20. 20. 20. 20. 20. 20.

B U L K

.. 5 101 102 103 104 105 201 202 203 204 205 301 302 303 304 305 5.0 5.0 5.0 5.0 5.0 20.0 20.0 20.0 20.0 20.0 25.0 25.0 25.0 25.0 25.0 0. 0. 20. 40. 60. 80. 100. 0. 20. 40. 60. 80. 100. 0. 20. 40. 60. 80. 100.

.. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

D A T A 6

0.

.. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

20.

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

8

E C H O 7

.. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.

8

0.

..

9

10.

..

10

+CRD

1246

1.244-5 0.0 1 2 1 4 10 10 1 .0025907 1 2.0 1.0 200 100 105 205 200 300 205 305

3 AUTO AUTO AUTO AUTO

1 1 1 1

1.0

100.

PW PW PW PW 0.0

YES YES YES YES

0.1

+PB100 $

123456 123456 123456 123456 +TABM1

10.3+6

200.0

10.3+6

ENDT

3.9+6

200.0

3.9+6

ENDT

1.244-5 100 103 300 303 100 103 300 303 100 103 300 303

200.0 0.001 0.001 0.001 0.001 50. 50. 50. 50. 150. 150. 150. 150.

1.244-5 101 104 301 304 101 104 301 304 101 104 301 304

ENDT 0.001 0.001 0.001 0.001 50. 50. 50. 50. 150. 150. 150. 150.

+TABM2 +TABM3 102 105 302 305 102 105 302 305 102 105 302 305

0.001 0.001 0.001 0.001 50. 50. 50. 50. 150. 150. 150. 150.

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Chapter 7

838485868788899091-

TEMP TEMP TEMP TEMP TEMPD TEMPD TEMPD TEMPD TEMPD ENDDATA TOTAL COUNT=

5 5 5 5 1 2 3 4 5

100 103 300 303 0. 0. 0. 0. 0.

200. 200. 200. 200.

101 104 301 304

200. 200. 200. 200.

102 105 302 305

200. 200. 200. 200.

92

Listing 7-48. Sorted Bulk Data for Example HA153A EXAMPLE HA153A: THREE BAR HEATED WING GEOMETRIC NONLINEAR STATICS AND HEAT TRANSFER ANALYSIS

PAGE

11

*** USER INFORMATION MESSAGE 4158 (DFMSA) ---STATISTICS FOR SPARSE DECOMPOSITION OF DATA BLOCK KLLRH FOLLOW NUMBER OF NEGATIVE TERMS ON FACTOR DIAGONAL = 1 MAXIMUM RATIO OF MATRIX DIAGONAL TO FACTOR DIAGONAL = 1.2E+17 AT ROW NUMBER *** USER WARNING MESSAGE 4698. STATISTICS FOR DECOMPOSITION OF MATRIX KLLRH . THE FOLLOWING DEGREES OF FREEDOM HAVE FACTOR DIAGONAL RATIOS GREATER THAN 1.00000E+07 OR HAVE NEGATIVE TERMS ON THE FACTOR DIAGONAL.

29

GRID POINT ID

DEGREE OF FREEDOM 105 301

MATRIX/FACTOR DIAGONAL RATIO

R2 T3

MATRIX DIAGONAL

1.20955E+17 2.61442E+16

7.48913E+06 2.97225E+04

N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 1 ITERATION TIME 0.34 SECONDS LOAD FACTOR 1.000 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 9.9000E+01 4.1903E-08 4.1903E-08 1.0000E-01 8.8379E-10 1.0000E+00 -1.0525E-09 -1.0525E-09 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 1 LOOPID 1 LOAD STEP 1.000 LOAD FACTOR 1.00000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 1 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 2 ITERATION TIME 0.33 SECONDS LOAD FACTOR 0.250 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 9.8992E+01 3.6350E-12 3.6347E-12 1.0000E-01 8.7635E-10 1.0000E+00 3.6759E-14 3.6759E-14 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 2 LOOPID 2 LOAD STEP 1.250 LOAD FACTOR 0.25000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 2 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 2 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.500 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 4.9498E+01 6.2422E-13 3.1210E-13 1.0000E-01 2.7370E-10 1.0000E+00 1.1959E-13 1.1959E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 2 LOOPID 3 LOAD STEP 1.500 LOAD FACTOR 0.50000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 3 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 2 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.750 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 3.2999E+01 3.5641E-12 1.1880E-12 1.0000E-01 1.0808E-09 1.0000E+00 -1.5573E-13 -1.5573E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 2 LOOPID 4 LOAD STEP 1.750 LOAD FACTOR 0.75000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) -

THE SOLUTION FOR LOOPID=

4 IS SAVED FOR RESTART

N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2,65 SECONDS SUBCASE 2 ITERATION TIME 0.34 SECONDS LOAD FACTOR 1.000 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 2.4750E+01 4.3176E-12 1.0794E-12 1.0000E-01 1.2466E-09 1.0000E+00 1.1400E-13 1.1400E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 2 LOOPID 5 LOAD STEP 2.000 LOAD FACTOR 1.00000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 5 IS SAVED FOR RESTART STIFFNESS UPDATE TIME ITERATION TIME

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N O N - L I N E A R 2.65 SECONDS 0.34 SECONDS

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I T E R A T I O N

M O D U L E

O U T P U T SUBCASE LOAD FACTOR

3 0.100

Flutter Analysis Sample Problems

- - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC ND MDV 1 1.6500E+01 8.0967E-13 1.3494E-13 1.0000E-01 1.1220E-09 1.0000E+00 -1.2858E-13 -1.2858E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 6 LOAD STEP 2.100 LOAD FACTOR 0.10000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 6 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.200 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 1.4143E+01 1.0795E-12 1.5422E-13 1.0000E-01 1.4963E-09 1.0000E+00 1.7127E-13 1.7127E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 7 LOAD STEP 2.200 LOAD FACTOR 0.20000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 7 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.36 SECONDS LOAD FACTOR 0.300 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 1.2375E+01 1.0795E-12 1.3493E-13 1.0000E-01 1.4966E-09 1.0000E+00 -1.7143E-13 -1.7143E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 8 LOAD STEP 2.300 LOAD FACTOR 0.30000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 8 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.35 SECONDS LOAD FACTOR 0.400 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 1.1000E+01 8.0966E-13 8.9962E-14 1.0000E-01 1.1225E-09 1.0000E+00 1.2857E-13 1.2857E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 9 LOAD STEP 2.400 LOAD FACTOR 0.40000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) - THE SOLUTION FOR LOOPID=

9 IS SAVED FOR RESTART

N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.35 SECONDS LOAD FACTOR 0.500 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 9.9000E+00 2.2060E-16 2.2060E-17 1.0000E-01 6.5303E-13 1.0000E+00 -5.9890E-16 -5.9890E-16 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 10 LOAD STEP 2.500 LOAD FACTOR 0.50000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 10 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.35 SECONDS LOAD FACTOR 0.600 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 9.0000E+00 8.0955E-13 7.3596E-14 1.0000E-01 1.1226E-09 1.0000E+00 -1.2896E-13 -1.2896E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 11 LOAD STEP 2.600 LOAD FACTOR 0.60000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 11 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.35 SECONDS LOAD FACTOR 0.700 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 8.2500E+00 1.0794E-12 8.9947E-14 1.0000E-01 1.4957E-09 1.0000E+00 1.7080E-13 1.7080E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 12 LOAD STEP 2.700 LOAD FACTOR 0.70000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 12 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.35 SECONDS LOAD FACTOR 0.800 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 7.6154E+00 1.0796E-12 8.3042E-14 1.0000E-01 1.4967E-09 1.0000E+00 -1.7142E-13 -1.7142E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 13 LOAD STEP 2.800 LOAD FACTOR 0.80000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 13 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 3 ITERATION TIME 0.35 SECONDS LOAD FACTOR 0.900 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 7.0714E+00 8.0969E-13 5.7835E-14 1.0000E-01 1.1234E-09 1.0000E+00 1.2811E-13 1.2811E-13 0 0 0 0 1

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*** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 14 LOAD STEP 2.900 LOAD FACTOR 0.90000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= STIFFNESS UPDATE TIME ITERATION TIME

N O N - L I N E A R 2.65 SECONDS 0.35 SECONDS

I T E R A T I O N

14 IS SAVED FOR RESTART

M O D U L E

O U T P U T SUBCASE LOAD FACTOR

3 1.000

- - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 6.6000E+00 1.1978E-16 7.9854E-18 1.0000E-01 1.6295E-12 1.0000E+00 -5.3236E-16 -5.3236E-16 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 3 LOOPID 15 LOAD STEP 3.000 LOAD FACTOR 1.00000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 15 IS SAVED FOR RESTART 1

N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.100 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 3.1935E+00 1.7409E-13 5.6158E-15 1.0000E-01 6.2418E-10 1.0000E+00 -1.4139E-13 -1.4139E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 16 LOAD STEP 3.100 LOAD FACTOR 0.10000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 16 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.200 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 3.0938E+00 1.7414E-13 5.4419E-15 1.0000E-01 6.2402E-10 1.0000E+00 1.4367E-13 1.4367E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 17 LOAD STEP 3.200 LOAD FACTOR 0.20000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 17 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.300 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 3.0000E+00 7.5423E-17 2.2855E-18 1.0000E-01 4.0268E-12 1.0000E+00 8.5177E-16 8.5177E-16 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 18 LOAD STEP 3.300 LOAD FACTOR 0.30000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 18 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.400 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 2.9118E+00 1.7405E-13 5.1190E-15 1.0000E-01 6.2355E-10 1.0000E+00 -1.4214E-13 -1.4214E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 19 LOAD STEP 3.400 LOAD FACTOR 0.40000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 19 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.34 SECONDS LOAD FACTOR 0.500 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 2.8286E+00 1.7418E-13 4.9767E-15 1.0000E-01 6.2483E-10 1.0000E+00 1.4388E-13 1.4388E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 20 LOAD STEP 3.500 LOAD FACTOR 0.50000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) - THE SOLUTION FOR LOOPID= 20 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T SUBCASE 4 STIFFNESS UPDATE TIME 2.65 SECONDS ITERATION TIME 0.33 SECONDS LOAD FACTOR 0.600 - - - LINE SEARCH DATA - - - - - CONVERGENCE FACTORS - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 2.7500E+00 8.8191E-14 2.4498E-15 1.0000E-01 1.9606E-10 1.0000E+00 1.1538E-13 1.1538E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 21 LOAD STEP 3.600 LOAD FACTOR 0.60000 THE SOLUTION FOR LOOPID= 21 IS SAVED FOR RESTART ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) O U T P U T N O N - L I N E A R I T E R A T I O N M O D U L E STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 LOAD FACTOR 0.700 ITERATION TIME 0.33 SECONDS - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV ITERATION EUI EPI EWI LAMBDA 1 2.6757E+00 1.7420E-13 4.7081E-15 1.0000E-01 6.2424E-10 1.0000E+00 -1.4441E-13 -1.4441E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED ***

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SUBID 4 LOOPID 22 LOAD STEP 3.700 LOAD FACTOR 0.70000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID=

22 IS SAVED FOR RESTART

N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.33 SECONDS LOAD FACTOR 0.800 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 2.6053E+00 1.7413E-13 4.5825E-15 1.0000E-01 6.2486E-10 1.0000E+00 1.4285E-13 1.4285E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 23 LOAD STEP 3.800 LOAD FACTOR 0.80000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 23 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.35 SECONDS LOAD FACTOR 0.900 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 2.5385E+00 9.6752E-17 2.4808E-18 1.0000E-01 1.0129E-12 1.0000E+00 -1.4551E-15 -1.4551E-15 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 24 LOAD STEP 3.900 LOAD FACTOR 0.90000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 24 IS SAVED FOR RESTART N O N - L I N E A R I T E R A T I O N M O D U L E O U T P U T STIFFNESS UPDATE TIME 2.65 SECONDS SUBCASE 4 ITERATION TIME 0.32 SECONDS LOAD FACTOR 1.000 - - - CONVERGENCE FACTORS - - - - - LINE SEARCH DATA - - ITERATION EUI EPI EWI LAMBDA DLMAG FACTOR E-FIRST E-FINAL NQNV NLS ENIC NDV MDV 1 2.4750E+00 1.7416E-13 4.3539E-15 1.0000E-01 6.2725E-10 1.0000E+00 -1.4459E-13 -1.4459E-13 0 0 0 0 1 *** USER INFORMATION MESSAGE 6186, *** SOLUTION HAS CONVERGED *** SUBID 4 LOOPID 25 LOAD STEP 4.000 LOAD FACTOR 1.00000 ^^^ DMAP INFORMATION MESSAGE 9005 (NLSCSH) THE SOLUTION FOR LOOPID= 25 IS SAVED FOR RESTART

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Listing 7-49. Output for Example HA153A N A S T R A N

F I L E

M A N A G E M E N T

S E C T I O N

E C H O

RESTART VERS=1 KEEP PAGE N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

3

E C H O

ID NXN, HA145KR TIME 5 $ CPU TIME IN MINUTES SOL 145 $ FLUTTER CEND EXAMPLE HA145KR: THREE BEAM HEATED WING GEOMETRIC NONLINEAR ANALYSIS AND FLUTTER

PAGE

C A S E CARD COUNT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

C O N T R O L

D E C K

E C H O

TITLE = EXAMPLE HA145KR: THREE BEAM HEATED WING SUBTI = GEOMETRIC NONLINEAR ANALYSIS AND FLUTTER ECHO = BOTH TEMP(INIT) = 1 $ INITIAL TEMPERATURE PARAM,NMLOOP,10 $ SPECIFIES LOOP IDENTIFICATION NUMBER IN SOL 153 FOR UPDATED STIFFNESS SPC = 1 $ ADDITIONAL CONSTRAINTS METHOD = 10 $ MGIV REAL EIGENVALUE METHOD FOR VIBES SVEC = ALL $ REQUEST VIBRATION MODES FMETHOD = 20 $ PK-FLUTTER METHOD $ OUTPUT(XYOUT) CSCALE 2.0 PLOTTER NASTRAN CURVELINESYMBOL = 6 YTTITLE = DAMPING G YBTITLE = FREQUENCY F HZ XTITLE = VELOCITY V (FT/S) XMIN = 0. XMAX = 300. YTMIN = -1.0 YTMAX = +1.0 YBMIN = 0. YBMAX = 10. XTGRID LINES = YES XBGRID LINES = YES YTGRID LINES = YES YBGRID LINES = YES UPPER TICS = -1 TRIGHT TICS = -1 BRIGHT TICS = -1 XYPLOT VG / 1(G,F) 2(G,F) $ BEGIN BULK

EXAMPLE HA145KR: THREE BEAM HEATED WING GEOMETRIC NONLINEAR ANALYSIS AND FLUTTER

PAGE

I N P U T . 1 $ $ $ PARAM $ $ $ $ SPC1 $ $ $ $ $ $ $ AERO $ $ $ $ $ $ $ $ $

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4

..

2

B U L K

..

3

..

D A T A 4

..

5

D E C K 5

..

6

E C H O ..

7

..

8

..

9

..

GRID POINT WEIGHT GENERATOR GRDPNT

100 $

ADDITIONAL CONSTRAINTS FOR CANTILEVER WING SID 1

C 35 *

*

G1 200 *

G2

AERODYNAMIC DATA

ETC *

*

*

THE AERO ENTRY GIVES BASIC AERODYNAMIC PARAMETERS FOR UNSTEADY AERODYNAMICS. ACSID

VELOCITY REFC 20.

RHOREF SYMXZ 1.1463-7 1

SYMXY 0

THE CAERO1 ENTRY IS USED FOR DOUBLET-LATTICE AND ZONA51 AERODYNAMICS. LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE USED TO PARTITION THE WING INTO AERODYNAMIC PANELS, THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER FOR NON-UNIFORMLY SPACED PANELS. IGID IS THE ID OF ITS ASSOCIATED INTERFERENCE GROUP. THE CONTINUATION ENTRY

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ $ $ $

10

.

Flutter Analysis Sample Problems

$ DEFINES POINTS 1 AND 4, THE ROOT CHORD AND TIP CHORD. $ $ THE BOXES FORMED BY THE GRID LINES WILL BE NUMBERED $ $ BEGINNING WITH EID SO CHOOSE A NUMBER THAT IS UNIQUE, $ $ AND IS GREATER THAN ALL STRUCTURAL GRID, SCALAR AND $ $ EXTRA POINT IDS. $ $ $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 1001 1000 8 4 1 +CAW $ ( FWD LEFT POINT ) CHORD ( FWD RIGHT POINT ) CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CAW -5. 0. 0. 40. -5. 100. 0. 40. $ $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1000 $ $ $ THE SPLINE2 ENTRY DEFINES A BEAM SPLINE FOR INTERPOLATING PANELS $ AND BODIES FOR AEROELASTIC PROBLEMS. $ $ $ EID CAERO ID1 ID2 SETG DZ DTOR CID SPLINE2 1001 1001 1001 1032 1100 0. 1.0 1 +SPL +SPL -1.0 -1.0 $ $ THE SET1 ENTRY DEFINES THE SETS OF STRUCTURAL GRID POINTS $ $ TO BE USED BY THE BEAM SPLINE FOR INTERPOLATION. $ $ $ $ SID G1 G2 G3 G4 SET1 1100 100 THRU 105 300 THRU 305 $ $ THE MKAERO1 ENTRY PROVIDES A TABLE OF MACH NUMBERS AND REDUCED $ FREQUENCIES FOR AERODYNAMIC MATRIX CALCULATION. $ $ M1 M2 ETC MKAERO1 0. +MK $ K1 K2 K3 K4 K5 ETC +MK .001 .1 .2 .5 1.0 $ $ $ $

$ * * VIBRATION SOLUTION PARAMETERS * * $ $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM; IN THIS CASE THE $ MODIFIED GIVENS METHOD. FOUR MODES ARE DESIRED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $ SID METHOD F1 F2 NO EIGR 10 MGIV 0. 25. 4 $ NORM G C +EGR MAX $ $ * * FLUTTER SOLUTION PARAMETERS * * $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE (DEFAULT IS 10-3). $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 20 PK 1 2 3 4 0.01 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NO.S $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ NEGATIVE VELOCITIES ARE SIGNALS TO COMPUTE AND PRINT EIGEN$ VECTORS. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 1.0 FLFACT 2 0. FLFACT 3 600. 1200. 1800. 1920. 2400. 3000. 3600. $ $ THE PARAMETER LMODES IS THE LOWEST NUMBER OF MODES TO BE $ USED IN A MODAL FORMULATION. $ PARAM LMODES 4 $ $ IN A FLUTTER ANALYSIS, THE VELOCITIES ARE DIVIDED BY THE $ PARAMETER VREF TO CONVERT UNITS OF VELOCITY IN THE OUTPUT $ FLUTTER SUMMARY TABLE. $ PARAM VREF 12.0 $ ENDDATA INPUT BULK DATA CARD COUNT = 110

$ $ $ $ $ $ $ +EGR $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH NO. VELOCITY $

Listing 7-50. Input Data for Example HA145KR

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EXAMPLE HA145KR: THREE BEAM HEATED WING GEOMETRIC NONLINEAR ANALYSIS AND FLUTTER S O R T E D CARD COUNT 1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071-

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B U L K

D A T A

. 1 .. 2 .. 3 .. 4 AERO 20. CAERO1 1001 1000 +CAW -5. 0. 0. CBEAM 101 100 100 CBEAM 102 100 101 CBEAM 103 100 102 CBEAM 104 100 103 CBEAM 105 100 104 CBEAM 201 100 200 CBEAM 202 100 201 CBEAM 203 100 202 CBEAM 204 100 203 CBEAM 205 100 204 CBEAM 301 100 300 CBEAM 302 100 301 CBEAM 303 100 302 CBEAM 304 100 303 CBEAM 305 100 304 CONM2 101 101 0 CONM2 102 102 0 CONM2 103 103 0 CONM2 104 104 0 CONM2 105 105 0 CONM2 201 201 0 CONM2 202 202 0 CONM2 203 203 0 CONM2 204 204 0 CONM2 205 205 0 CONM2 301 301 0 CONM2 302 302 0 CONM2 303 303 0 CONM2 304 304 0 CONM2 305 305 0 CORD2R 1 0 20. +CRD 30. 0. 0. EIGR 10 MGIV 0. +EGR MAX FLFACT 1 1.0 FLFACT 2 0. FLFACT 3 600. 1200. FLUTTER 20 PK 1 GRID 100 0. GRID 101 0. GRID 102 0. GRID 103 0. GRID 104 0. GRID 105 0. GRID 200 10. GRID 201 10. GRID 202 10. GRID 203 10. GRID 204 10. GRID 205 10. GRID 300 20. GRID 301 20. GRID 302 20. GRID 303 20. GRID 304 20. GRID 305 20. MAT1 1 10.3+6 3.9+6 MATT1 1 1 2 MKAERO1 0. +MK .001 .1 .2 NLPARM 10 1 NLPARM 20 4 NLPARM 30 10 NLPARM 40 10 PAERO1 1000 PARAM GRDPNT 100 PARAM LGDISP 1 LMODES

4 PARAM PARAM PBEAM +PB100 RBAR RBAR RBAR RBAR SET1 SPC1 SPLINE2 +SPL TABLEM1 +TABM1

VREF WTMASS 100 NO 100 105 200 205 1100 1 1001 -1.0 1 0.0

NX Nastran Aeroelastic Analysis User’s Guide

12.0 .0025907 1 2.0 1.0 200 100 105 205 200 300 205 305 100 THRU 35 200 1001 1001 -1.0

8

E C H O .. 5 .. 6 1.1463-71 8 4 40. -5. 101 0. 102 0. 103 0. 104 0. 105 0. 201 0. 202 0. 203 0. 204 0. 205 0. 301 0. 302 0. 303 0. 304 0. 305 0. 5.0 5.0 5.0 5.0 5.0 20.0 20.0 20.0 20.0 20.0 25.0 25.0 25.0 25.0 25.0 0. 0. 25.

1800. 2 0. 20. 40. 60. 80. 100. 0. 20. 40. 60. 80. 100. 0. 20. 40. 60. 80. 100.

.. 0

7

..

100. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.

20.

0.

8

..

9

2400.

10

1 40.

+CAW

10.

+CRD

4

1920. 3 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

..

+EGR

3000. 4

3600. 0.01

DENSITY MACH NO. VELOCITY

1246

1.244-5 0.0 3 +MK .5 AUTO AUTO AUTO AUTO

1.0 1 1 1 1

1.0

100.

0.0

0.1

123456 123456 123456 123456 105

300

THRU

305

1032

1100

0.

1.0

10.3+6

ENDT

PW PW PW PW

YES YES YES YES

+PB100 $

1

+SPL +TABM1

10.3+6

200.0

.

Flutter Analysis Sample Problems

8687888990919293949596979899100101102103104105106107108109110-

TABLEM1 +TABM2 TABLEM1 +TABM3 TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMP TEMPD TEMPD TEMPD TEMPD TEMPD ENDDATA TOTAL COUNT=

2 0.0 3 0.0 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 1 2 3 4 5

+TABM2 3.9+6

200.0

3.9+6

ENDT

1.244-5 100 103 300 303 100 103 300 303 100 103 300 303 100 103 300 303 0. 0. 0. 0. 0.

200.0 0.001 0.001 0.001 0.001 50. 50. 50. 50. 150. 150. 150. 150. 200. 200. 200. 200.

1.244-5 101 104 301 304 101 104 301 304 101 104 301 304 101 104 301 304

ENDT 0.001 0.001 0.001 0.001 50. 50. 50. 50. 150. 150. 150. 150. 200. 200. 200. 200.

+TABM3 102 105 302 305 102 105 302 305 102 105 302 305 102 105 302 305

0.001 0.001 0.001 0.001 50. 50. 50. 50. 150. 150. 150. 150. 200. 200. 200. 200.

111

Listing 7-51. Sorted Bulk Data for Example HA145KR EXAMPLE HA145KR: THREE BEAM HEATED WING GEOMETRIC NONLINEAR ANALYSIS AND FLUTTER O U T P U T

* * * * * *

PAGE

F R O M

G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 100 M O 2.500000E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.500000E+04 0.000000E+00 2.500000E+02 0.000000E+00 0.000000E+00 0.000000E+00 3.500000E+03 0.000000E+00 0.000000E+00 2.500000E+02 1.500000E+04 -3.500000E+03 0.000000E+00 0.000000E+00 0.000000E+00 1.500000E+04 1.100000E+06 -2.100000E+05 0.000000E+00 0.000000E+00 0.000000E+00 -3.500000E+03 -2.100000E+05 6.000000E+04 0.000000E+00 -1.500000E+04 3.500000E+03 0.000000E+00 0.000000E+00 0.000000E+00 1.160000E+06 S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 2.500000E+02 0.000000E+00 6.000000E+01 0.000000E+00 Y 2.500000E+02 1.400000E+01 0.000000E+00 0.000000E+00 Z 2.500000E+02 1.400000E+01 6.000000E+01 0.000000E+00 I(S) * 2.000001E+05 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.100000E+04 0.000000E+00 * * 0.000000E+00 0.000000E+00 2.110001E+05 * I(Q) * 2.000001E+05 * * 1.100000E+04 * * 2.110001E+05 * Q * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * R E A L

MODE NO. 1 2 3 4 5 6 7 8 9 10

EXTRACTION ORDER 28 29 30 31 32 1 33 34 35 39

3.128240E+02 1.259820E+03 1.219031E+04 2.912658E+04 5.821704E+04 6.004134E+04 1.163347E+05 1.703640E+05 3.409595E+05 5.294054E+05

EIGENVALUE = CYCLES =

3.128240E+02 2.814946E+00

POINT ID. 101 102 103 104 105 201 202

EIGENVALUE

TYPE G G G G G G G

12

RADIANS 1.768683E+01 3.549394E+01 1.104097E+02 1.706651E+02 2.412821E+02 2.450333E+02 3.410787E+02 4.127518E+02 5.839174E+02 7.276025E+02 R E A L

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0

* * * * * *

E I G E N V A L U E S CYCLES GENERALIZED GENERALIZED MASS STIFFNESS 2.814946E+00 3.176117E-01 9.935656E+01 5.649036E+00 5.702050E-02 7.183556E+01 1.757225E+01 2.102801E-01 2.563379E+03 2.716220E+01 1.428661E-01 4.161202E+03 3.840124E+01 0.0 0.0 3.899827E+01 0.0 0.0 5.428436E+01 0.0 0.0 6.569149E+01 0.0 0.0 9.293333E+01 0.0 0.0 1.158015E+02 0.0 0.0

E I G E N V E C T O R T3 7.296664E-03 5.967955E-02 1.893437E-01 4.037478E-01 6.836064E-01 8.162597E-02 2.609250E-01

N O . R1 1.147712E-03 4.378497E-03 8.654088E-03 1.261274E-02 1.502141E-02 7.159712E-03 1.033114E-02

1 R2 -6.525782E-03 -1.305156E-02 -1.957735E-02 -2.610313E-02 -3.262891E-02 -6.525782E-03 -1.305156E-02

NX Nastran Aeroelastic Analysis User’s Guide

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0

7-197

Chapter 7

203 204 301 302 303 304

Flutter Analysis Sample Problems

G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0

EIGENVALUE = 1.259820E+03 CYCLES = 5.649036E+00 POINT ID. TYPE T1 101 G 0.0 102 G 0.0 103 G 0.0 104 G 0.0 105 G 0.0 201 G 0.0 202 G 0.0 203 G 0.0 204 G 0.0 301 G 0.0 302 G 0.0 303 G 0.0 304 G 0.0 EIGENVALUE = CYCLES = POINT ID. 101 102 103 104 105 201 202 203 204 301 302 303 304 EIGENVALUE = CYCLES =

1.219031E+04 1.757225E+01 TYPE G G G G G G G G G G G G G

2.912658E+04 2.716220E+01

1.184416E-02 1.300315E-02 7.616369E-03 1.434532E-02 1.815709E-02 1.817642E-02

R E A L T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

E I G E N V E C T O T3 R1 7.621557E-02 2.826044E-01 5.571611E-01 8.194914E-01 1.000000E+00 3.314885E-02 1.045081E-01 1.906229E-01 2.840572E-01 -5.506493E-02 -1.724937E-01 -2.733315E-01 -2.944551E-01

R E A L

E I G E N V E C T O R

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

4.842834E-01 7.324311E-01 7.558579E-02 2.987881E-01 6.301132E-01 1.000000E+00

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R E A L

T3 2.114693E-01 6.138327E-01 8.594995E-01 7.216847E-01 1.925080E-01 9.502361E-02 2.414319E-01 2.899759E-01 1.397008E-01 4.305721E-01 1.000000E+00 9.029533E-01 8.058599E-02

R

N O . R2 7.430675E-03 1.264947E-02 1.408163E-02 1.152371E-02 6.226471E-03 2.887687E-03 4.052567E-03 4.492481E-03 4.908875E-03 -4.938958E-03 -6.086522E-03 -3.423079E-03 1.503537E-03 N O .

R1 1.829058E-02 1.876361E-02 3.729860E-03 -1.755753E-02 -3.350382E-02 7.499272E-03 5.829746E-03 -1.828819E-03 -1.396354E-02 3.344695E-02 1.578493E-02 -2.593060E-02 -4.982566E-02

E I G E N V E C T O R

N O .

-1.957735E-02 -2.610313E-02 -6.525782E-03 -1.305156E-02 -1.957735E-02 -2.610313E-02 2 R3 1.214537E-02 2.429073E-02 3.643610E-02 4.858146E-02 6.072683E-02 1.214537E-02 2.429073E-02 3.643610E-02 4.858146E-02 1.214537E-02 2.429073E-02 3.643610E-02 4.858146E-02

R2 1.016710E-02 2.033420E-02 3.050130E-02 4.066840E-02 5.083550E-02 1.016710E-02 2.033420E-02 3.050130E-02 4.066840E-02 1.016710E-02 2.033420E-02 3.050130E-02 4.066840E-02

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

4

FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.0000 DENSITY RATIO = 1.0000E+00 METHOD = PK FREQUENCY COMPLEX EIGENVALUE KFREQ 1./KFREQ VELOCITY DAMPING 0.2951 3.3883512E+00 5.0000000E+01 -3.3877235E-02 2.8182724E+00 -2.9994443E-01 1.7707727E+01 0.1519 6.5851388E+00 1.0000000E+02 -7.9980932E-02 2.9002566E+00 -7.2874027E-01 1.8222851E+01 0.1128 8.8682280E+00 1.5000000E+02 -2.0572214E-01 3.2303958E+00 -2.0877893E+00 2.0297176E+01 0.1096 9.1269636E+00 1.6000000E+02 -3.5693970E-01 3.3480740E+00 -3.7543933E+00 2.1036570E+01 0.0725 1.3790782E+01 2.0000000E+02 -1.0542395E+00 2.7697618E+00 -9.1734266E+00 1.7402927E+01 0.0000 9.9999996E+24 2.5000000E+02 -8.3181366E-02 0.0000000E+00 -8.6484489E+00 0.0000000E+00 0.0000 9.9999996E+24 3.0000000E+02 1.2051423E-01 0.0000000E+00 1.5035982E+01 0.0000000E+00 DENSITY RATIO = 1.0000E+00 METHOD = PK POINT = 2 MACH NUMBER = 0.0000 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.5764 1.7347887E+00 5.0000000E+01 -2.3110120E-02 5.5045872E+00 -3.9964724E-01 3.4586342E+01 0.2665 3.7522483E+00 1.0000000E+02 -5.5297688E-02 5.0899062E+00 -8.8423282E-01 3.1980825E+01 0.1435 6.9690208E+00 1.5000000E+02 -5.0988071E-02 4.1107478E+00 -6.5847504E-01 2.5828592E+01 0.1243 8.0450697E+00 1.6000000E+02 5.2899200E-02 3.7983196E+00 6.3123417E-01 2.3865547E+01

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

3

POINT ID. TYPE T1 T2 T3 R1 R2 101 G 0.0 0.0 -3.828819E-02 -3.670634E-03 -6.925469E-03 102 G 0.0 0.0 -1.390392E-01 -6.163489E-03 -1.385094E-02 103 G 0.0 0.0 -2.800803E-01 -7.938215E-03 -2.077641E-02 104 G 0.0 0.0 -4.677928E-01 -1.141089E-02 -2.770188E-02 105 G 0.0 0.0 -7.755890E-01 -2.079395E-02 -3.462734E-02 201 G 0.0 0.0 4.484498E-01 3.311279E-02 -6.925469E-03 202 G 0.0 0.0 1.000000E+00 1.539818E-02 -1.385094E-02 203 G 0.0 0.0 9.462705E-01 -2.081512E-02 -2.077641E-02 204 G 0.0 0.0 2.712620E-01 -4.076002E-02 -2.770188E-02 301 G 0.0 0.0 -1.499004E-01 -1.053164E-02 -6.925469E-03 302 G 0.0 0.0 -2.854689E-01 -5.220476E-04 -1.385094E-02 303 G 0.0 0.0 -1.589953E-01 1.127049E-02 -2.077641E-02 304 G 0.0 0.0 4.717688E-02 5.187527E-03 -2.770188E-02 A ZERO FREQUENCY ROOT HAS EMERGED. WHEN THE MACH NO., DENSITY AND VELOCITY ARE COMPATIBLE IT MAY BE INTERPRETED TWO WAYS DEPENDING ON THE SIGN OF THE REAL PART: 1. (-) A MODE IS CRITICALLY DAMPED, OR, 2. (+) THE SYSTEM IS DIVERGING. ONLY THE MOST CRITICAL ( I.E., MOST POSITIVE REAL ROOTS ) ARE PRINTED IN THE FLUTTER SUMMARY. FOR INFORMATIONAL PURPOSES, THE REMAINING REAL ROOTS ARE PRINTED HERE. LESS CRITICAL REAL ROOTS FOR LOOP 6 MACH VELOCITY DENSITY 0.00000E+00 3.00000E+03 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -1.6773781E-01 -1.7439867E+01 0.0000000E+00 0.0000 -8.3181366E-02 -8.6484489E+00 0.0000000E+00 LESS CRITICAL REAL ROOTS FOR LOOP 7 MACH VELOCITY DENSITY 0.00000E+00 3.60000E+03 1.00000E+00 KFREQ DAMPING COMPLEX EIGENVALUE 0.0000 -2.9823565E-01 -3.7209431E+01 0.0000000E+00 0.0000 1.2051423E-01 1.5035982E+01 0.0000000E+00

7-198

0.0 0.0 0.0 0.0 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Flutter Analysis Sample Problems

0.0881 0.0567 0.0366 POINT = KFREQ 1.8269 0.9106 0.6069 0.5690 0.4553 0.3644 0.3038 POINT = KFREQ 2.8423 1.4202 0.9463 0.8871 0.7094 0.5672 0.4723

1.1355899E+01 1.7638226E+01 2.7339989E+01 3 MACH 1./KFREQ 5.4737139E-01 1.0982338E+00 1.6478065E+00 1.7575625E+00 2.1962969E+00 2.7441893E+00 3.2911725E+00 4 MACH 1./KFREQ 3.5182920E-01 7.0411587E-01 1.0567598E+00 1.1273248E+00 1.4097354E+00 1.7631882E+00 2.1172702E+00

2.0000000E+02 4.5322740E-01 2.5000000E+02 8.1069690E-01 3.0000000E+02 5.2598155E-01 NUMBER = 0.0000 DENSITY VELOCITY DAMPING 5.0000000E+01 -3.7979512E-03 1.0000000E+02 -1.1912663E-02 1.5000000E+02 -1.9875834E-02 1.6000000E+02 -2.1441333E-02 2.0000000E+02 -2.7598601E-02 2.5000000E+02 -3.5089836E-02 3.0000000E+02 -4.2351328E-02 NUMBER = 0.0000 DENSITY VELOCITY DAMPING 5.0000000E+01 -4.7117035E-05 1.0000000E+02 -5.5711548E-04 1.5000000E+02 -1.0402703E-03 1.6000000E+02 -1.1345085E-03 2.0000000E+02 -1.5126284E-03 2.5000000E+02 -1.9833411E-03 3.0000000E+02 -2.4335631E-03

3.3636425E+00 4.7893424E+00 2.1134390E+01 2.7069888E+00 6.8943744E+00 1.7008512E+01 2.0956767E+00 3.4629376E+00 1.3167525E+01 RATIO = 1.0000E+00 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 1.7445736E+01 -2.0815581E-01 1.0961479E+02 1.7390278E+01 -6.5082657E-01 1.0926634E+02 1.7385468E+01 -1.0855795E+00 1.0923612E+02 1.7386435E+01 -1.1711491E+00 1.0924220E+02 1.7391630E+01 -1.5079162E+00 1.0927483E+02 1.7399120E+01 -1.9180437E+00 1.0932190E+02 1.7408926E+01 -2.3162684E+00 1.0938351E+02 RATIO = 1.0000E+00 METHOD = PK FREQUENCY COMPLEX EIGENVALUE 2.7141851E+01 -4.0176059E-03 1.7053729E+02 2.7124218E+01 -4.7473621E-02 1.7042650E+02 2.7109175E+01 -8.8595644E-02 1.7033197E+02 2.7106424E+01 -9.6611738E-02 1.7031470E+02 2.7095285E+01 -1.2875849E-01 1.7024471E+02 2.7079626E+01 -1.6872910E-01 1.7014632E+02 2.7061155E+01 -2.0688967E-01 1.7003026E+02

NX Nastran Aeroelastic Analysis User’s Guide

7-199

Chapter 7

Flutter Analysis Sample Problems

Listing 7-52. Output for Example HA145KR

7-200

NX Nastran Aeroelastic Analysis User’s Guide

Chapter

8 Aeroelastic Design Sensitivities and Optimization

•

Overview

•

Aeroelastic Optimization of FSW Airplane (Example HA200A)

•

Aeroelastic Optimization of FSW Airplane (Example HA200B)

NX Nastran Aeroelastic Analysis User’s Guide

8-1

Chapter 8

8.1

Aeroelastic Design Sensitivities and Optimization

Overview

Aeroelastic design sensitivity and optimization are available in Solution 200. These capabilities are based on static aeroelastic analysis, Solution 144, and flutter analysis, Solution 145. Optimization of aeroelastic characteristics can therefore be combined with the other optimization features of Solution 200. Flight vehicles can then be designed optimally for aeroelastic loads, flying qualities, and flutter as well as for strength, vibration frequencies, and buckling characteristics. The theory, algorithms, and applications for aeroelastic optimization have been published by Neill, Johnson, and Canfield (1987), Johnson and Venkayya (1988), and Johnson and Reymond (1991). The specification of an aeroelastic design requires the specification of the response quantities that are to be included in the design process and the determination of these responses. NX Nastran permits the specification of a number of responses for static analysis including displacements, stresses, strains, and forces, and these capabilities are available in static aeroelasticity. The response quantities for aeroelastic design include: •

Trim variables in a static aeroelastic response. These allow the user to design for limits imposed on, e.g., the trim angle of attack or control surface travel.

•

Stability derivatives. These allow the user to specify limits on, e.g., the rolling effectiveness of an aileron.

•

Flutter damping level. This assures flutter stability without a requirement to determine the actual flutter speed.

Once a quantity is identified as a response, it can be used directly in a constraint condition; in combination with other responses and/or design variables in a constraint condition; as an objective function directly; or in combination with other responses and/or design variables in the objective function. This capability of combining responses and design variables is a unique innovation in NX Nastran. The user interface for the design sensitivity and optimization capability builds on the existing interface for aeroelastic analysis as detailed in the this guide and on the interface for design optimization described in the NX Nastran Design Sensitivity and Optimization User’s Guide . The integrated capability contains some aspects of the user interface that are unique to aeroelasticity.

Case Control Two aeroelastic analysis types are available for the ANALYSIS command: ANALYSIS = SAERO ANALYSIS = FLUTTER

As the names imply, the ANALYSIS commands refer to static aeroelasticity and flutter analysis, respectively.

Bulk Data Three response types for the DRESP1 entry are unique to aeroelasticity: TRIM, STABDER, and FLUTTER. The TRIM response is used to identify an aerodynamic extra point that the user wants to study. This is done by identifying the variable (AESURF or AESTAT ID) in the ATTA field of the DRESP1 entry.

8-2

NX Nastran Aeroelastic Analysis User’s Guide

Aeroelastic Design Sensitivities and Optimization

The STABDER response is used to identify a stability derivative. In addition to the aerodynamic variable, the user must identify the component (1 through 6) and if a restrained or unrestrained derivative is of interest. The FLUTTER response allows the user selectively to identify damping responses that are to be studied. Attributes that are available for the response are mode number, Mach number, density, and velocity. As with the input, the output is a combination of the existing aeroelastic and design optimization output. Perhaps the most valuable output available is the sensitivity matrix. This provides a quantitative assessment of how the responses change due to changes in the design variables [in the NX Nastran Design Sensitivity and Optimization User’s Guide . The effect of design changes on aeroelastic responses is often nonintuitive, even for the most experienced analyst. Therefore, the sensitivity matrix can provide substantial guidance when a design change is required or contemplated.

8.2

Aeroelastic Optimization of FSW Airplane (Example HA200A)

This example illustrates the design sensitivity features of SOLution 200 when aeroelastic constraints are included along with stress and deflection constraints. A final Example HA200B illustrates optimization. The Bulk Data sets are almost the same in the two examples with the differences noted at the end of the Bulk Data description contained in this section. These examples are based on the two-sided FSW airplane model of Example HA144E . The two-sided model, shown in Figure 8-1, is chosen since some of the design flight conditions are asymmetric maneuvers. The optimization is intended to minimize the structural weight of the wings and fin, as indicated in Figure 8-2, with constraints on tensile and compressive stresses and wing tip twist during four subsonic and one supersonic maneuver. Additional constraints are imposed on subsonic and supersonic rolling effectiveness and on the flutter speed. Note that the two-sided model can flutter either symmetrically or antisymmetrically but that the mode of flutter is not a concern with the two-sided model. The loading conditions are different from those analyzed in Example HA144E. The five flight conditions considered here are: (1) a subsonic pullout at m = 0.9 and nz = 6.0 Gs; (2) a supersonic pullout at m = 1.2 and nz = 4.0 Gs; (3) a subsonic steady rolling pullout at m = 0.9 and nz = 4.8 Gs; (4) a subsonic abrupt rolling pullout at m = 0.0 and nz = 4.8 Gs; and, (5) an entry into a snap-roll from level flight at m = 0.45. For the sensitivity task of this example, the goal is to see what effect the CBAR elements, which represent the wing and fin structures, have on selected responses. In developing the design model, the first task is the definition of the design variables. In this case, it is first necessary to define independent design variables that have no explicit physical meaning except that the physical properties of the CBARs are linearly related to these design variables:

Equation 8-1.

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where A, I1 , I2 , and J are the area, cross-section area moments of inertia, and effective torsional moment of inertia, respectively, of the i-th CBAR element. The design variable representing the inboard wing CBAR is x1 and the vector of coefficients is chosen to reproduce the physical properties in Example HA144E when x1 = 1.0. The same relationship is used to design the outboard wing CBAR using a second design variable x2 , and to design the fin CBAR using a third design variable x3 . The design of mass balances is another possible option for flutter prevention that is not considered in this example. Anticipating the design optimization task of the next section , a design objective of weight is defined in this section even though it is not required for the sensitivity task. The baseline input Example HA144E presents a problem since all of its mass data are input using CONM2 entries. Currently, optimization in NX Nastran doesn’t treat mass values input on the CONM2 entry as design parameters. To avoid this problem, the CONM2 values for the wing and fin are reduced to half of their values in Example HA144E values, and these values, may be regarded here as nonstructural mass; the nonstructural weight of each wing is now 1000 lbs, and the nonstructural weight of the fin is 50 lbs. The structural mass is then calculated by means of the CBAR cross-sectional area, A above, and its density on the MAT1 entry. For the wing, MAT1 2, is assigned a density of 38.49002 lbs/ft3, and for the fin, MAT1 3, is assigned a density of 5.773503 lbs/ft3. These values of density lead to a structural wing weight of 1000.0 lbs on each side and a structural fin weight of 50.0 lbs when the design variables x1 = x2 = x3 = 1.0. There is a slight aft shift of the center of gravity from its location in Example HA144E as a consequence of this division in weight between structural and nonstructural portions (it has moved from 2.276 ft to 2.514 ft aft of GRID 90, i.e., 2.38% of the mean aerodynamic chord).

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Figure 8-1. Aerodynamic Model of Example HA144E The final step in the design model specification relates to constraint specifications. This example illustrates typical preliminary design considerations: strength of the aeroelastic airframe in high load factor maneuvers, stiffness in maneuvering for good aerodynamic performance, stiffness for adequate roll control, and stiffness for flutter prevention. These design considerations are each expressed as constraints. Strength is prescribed as a constraint on stresses. Stiffness in maneuvering is prescribed as a constraint on the twist at the wing tips. Stiffness for roll control is prescribed as a constraint on the rolling helix angle per unit aileron deflection. Finally, stiffness for flutter prevention is prescribed by constraints on the aeroelastic system damping at selected speeds in the flight envelope, including margins of safety on velocity. The foregoing design considerations are implemented in NX Nastran using the Bulk Data of Listing 8-1.

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Figure 8-2. Structural Model of HA200A and HA200B The listing begins with an INCLUDE entry for data set FSW_TWO.DAT, which is given in Listing 6-19. This is followed by those parts of the structural model that differ from the Example HA144E description due to design considerations. For example, separate PIDs are now required on the CBAR Bulk Data entry so that the associated properties can be independently designed. The CONM2 and MAT1 entries have been modified to split the aircraft weight into nonstructural and structural portions, as described above. Five TRIM Bulk Data entries define the five maneuvering flight conditions that are used to design the structure, with the remaining static aeroelastic inputs provided in the INCLUDE file. Input for the flutter analysis is in addition to that provided in Section and is therefore input in its entirety. A subsonic and a supersonic flutter analysis are specified with two separate FLUTTER Bulk Data entries and their corresponding FLFACT entries. PARAM,LMODES retains 12 modes in the flutter analysis.

Design Variables The remainder of the input is for the design model and begins with a definition of the design variables on the DESVAR Bulk Data entries. Design variables 10, 20, and 30 control the properties on the inboard wing, outboard wing, and fin bars, respectively. Note that the initial fin design variable is 10 percent of the wing design variables, reflecting the designer’s insight that the fin structure will require significantly less strength than that of the wing. The DVPREL1

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entries indicate the linear relationships between the PBAR properties and the design variables as specified in Eq. 8-1.

Responses and Constraints Responses required in the automated design task are identified using DRESP1 and DRESP2 Bulk Data entries. This example has considerable generality in that it contains DRESP1 entries for WEIGHT, STRESS, DISPlacement, STABDER (stability derivatives), and FLUTTER responses that are either constrained directly (i.e., specified to be within limiting values), used in specifying the design objective (in this case it is to minimize the weight), or included in synthetic responses. The STRESS responses invoke the minimum and maximum stresses at end A (the end at the grid point mentioned first on the CBAR entry) of the bars that model the wing structure. DCONSTR entries then constrain these stresses to fall between −57.6 and 72.0 × 106 psf. The ability to limit the structural deformations is demonstrated in this example by first identifying rotational displacements at the two wing tips and at the vehicle centerline. The difference between the tip and root displacement is a measure of the aeroelastic deformation, and the difference is effected using a DRESP2 entry that identifies the associated responses and a DEQATN entry that provides that equation the takes the difference. DCONSTR entries are used to limit the response to +1.0 degrees subsonically and +0.5 degrees supersonically. DCONADD entries adds these displacement constraints (plus a roll effectiveness constraint described next) to the already defined stress constraints. DCONADD 1 refers to the subsonic subcase while DCONADD 2 refers to the supersonic subcase. The roll performance specification frequently determines the stiffness of an aircraft wing. This requirement is demonstrated in this example by selecting two stability responses, Clp and Clδa, on DRESP1 entries and then defining an equation using the DRESP2 and DEQATN entries to specify a roll performance criterion

DCONSTR entries are used to provide a lower limit of 0.60 for this criterion subsonically and 0.43 supersonically.

Flutter Responses Two sets of DRESP1 entries include subsonic and supersonic flutter responses in the design task. DRESP1 1 selects modal damping responses at the analysis density and Mach number, but only at three critical velocities: 1000.0, 1300.0 and 1500.0 ft/sec. These responses must all be available from the analysis. A SET1 entry invoked by the DRESP1 entry restricts the responses to the fourth through the seventh modes since the first three modes represent rigid body behavior that should not impact the structural design. DRESP1 11 performs the same function for the supersonic flutter analysis. DRESP2 entries convert the damping value to a new response of the form

This form offsets the response from zero and scales it (see Aeroelastic Modeling) so that the constraint imposed on the DCONSTR entry specifies that this response should be less than -0.3, which is equivalent to saying that the damping should be negative.

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Optimization Parameters The input file concludes with several entries that define parameters for the optimization and sensitivity task. The DOPTPRM entry is used primarily in optimization and in this case limits the number of design cycles to 25, requests a comprehensive printout of design details following each design cycle, and overrides the default finite difference step size DELB to give greater numerical accuracy when computing sensitivity values. PARAM,CDIF,YES invokes the more accurate central differencing technique for this example. PARAM,NASPRT,2 requests standard NX Nastran output after every other design cycle. Even for this small model, this request results in extensive output so PARAM,NASPRT,2 must be used with caution. PARAM,OPTEXIT,4 specifies that sensitivity results for the baseline design are to be printed without proceeding on to the optimization phase. Two DSCREEN entries complete the Bulk Data input and, for this example, force the retention of all the constraints and therefore produce the sensitivities of all DRESP1 and DRESP2 responses. The DSCREEN, PARAM,OPEXIT and PARAM,CDIF entries are removed for the optimization example of the next section, and DELB is returned to its default value.

Case Control Commands The Case Control Section begins with the TITLE and SUBTITLE commands. The ECHO = BOTH command echoes the input Bulk Data, including the "INCLUDEd" files in unsorted and sorted formats. A single boundary condition is used for all the analyses and is selected by the SPC = 1 command. The DESOBJ = 10 request invokes the Bulk Data DRESP1 entry with ID = 10 as the objective. This is the WEIGHT response. Request for output of all the DISPlacments, STRESSes, element FORCEs, AEROdynamic Forces, and Aerodynamic PRESsures is made above the subcase level, which means that the requests apply to all the subcases until they are cancelled in subcases 6 and 7. Seven subcases follow, with each subcase defined by a unique LABEL. The first five subcases are static aeroelastic trim analyses for differing maneuvering conditions. The ANALYSIS = SAERO command is required in these subcases to identify which of the multidisciplinary analyses applies for the subcase. The TRIM command selects a TRIM Bulk Data entry that specifies the maneuver. For the first subcase, the DESSUB = 1 command invokes the DCONADD Bulk Data entry with DCID = 1 which, in turn, invokes limits on the element stresses, the aeroelastic deformation, and the roll performance for the subsonic flight condition. The second subcase invokes a similar set of constraints for the supersonic flight condition. The remaining three static aeroelastic subcases only constrain the stress responses. The final two subcases involve the flutter design tasks, as indicated by the ANALYSIS = FLUTTER commands. The sixth subcase, and the first flutter subcase, includes a DISP = 10 request along with PARAM,OPPHIPA set to 1 so as to obtain normal mode displacements at the grid points. SET 10 is used to limit to the displacement output to the aerodynamic box centers and not the box corners. The remaining output requests are set to NONE to avoid output of these data for eigenvectors. The METHOD and FMETHOD entries are familiar from Flutter Analysis Sample Problems and invoke the normal modes and flutter analysis, respectively. The DESSUB = 6 command invokes a DCONSTR entry that places limits on the subsonic flutter responses. The DESSUB = 7 command of the final subcase performs the same function for the supersonic flutter analysis. The Executive Control Section begins with the identification ID NXN, HA200A, which denotes the problem in the Test Problem Library. TIME 30 limits the CPU time to 30 minutes, and SOL 200 invokes the design sensitivity and optimization solution sequence.

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Output The input data and the sorted Bulk Data entries are shown in Listing 8-1 and Listing 8-2. Selected output is in Listing 8-3 and only includes those aspects of the output that are unique to design sensitivity since the lengthy remaining output has been covered by the examples in Static Aeroelastic Analysis Sample Problems and Flutter Analysis Sample Problems. An exception to this is that the subsonic flutter results are shown to assist in the discussion of the flutter sensitivity results. The first table shown in Listing 8-3 contains information on the properties that are being designed. In the table, the "ANALYSIS VALUE" refers to the values of the properties as they are input on the property entries (PBAR in this example) while the "DESIGN MODEL" refers to the values of the properties developed from the design model input on the DESVAR and DVPREL1 entries. These two values can differ, and the user is warned if they do. In this example, the properties on the fin for the design model are a tenth of the analysis properties. This was the user’s intent so the warning can be ignored. Limits on the property values are also displayed in this table. The flutter summary for the subsonic flutter analysis is shown next. It is seen that flutter occurs for point 6 between 1300.0 and 1400.0 ft/sec.

The Design Sensitivity Information NX Nastran produces a sensitivity matrix with rows that correspond to the design variables and the columns correspond to the responses. The next series of tables in Listing 8-3 identifies the columns of this matrix by listing attributes of the particular response. For example, the first response is the weight while the next three are the displacements identified on DRESP1 entries for the first subcase. This identification continues though all 115 columns of design sensitivity matrix.

The Design Sensitivity Matrix The design sensitivity matrix (MATRIX DSCM2) is then listed. Using the first column as an example, it shows that the sensitivity of the weight with respect to the three design variables is 666.68, 13333.30 and 49.9983, respectively. While this particular sensitivity is predictable and could be calculated by hand, it can be seen that other sensitivity values are not available without a computer analysis. For example, column 70 is the sensitivity of the sixth flutter mode at a velocity of 1500.0 ft/sec and a Mach number of 0.9. From the Flutter Summary, it is seen that the flutter damping at this point is positive (g = 0.4382). The sensitivities of this response with respect to the three design variables are -0.5986, -0.2431 and -0.0510, respectively. This provides the analyst/designer with useful information on the tradeoffs required to improve the flutter performance of the vehicle. In this particular case, it is seen that increasing the first design variable is the most cost effective strategy, i.e., achieves the most increase in flutter damping for a unit change in weight.

Design Variable Identification The final information in Listing 8-3 is the design cycle history and the design variable history. This information is most useful in an optimization task, but it also is useful for the sensitivity analysis since it identifies the design variable labels and values and therefore allows the user to quickly interpret the physical meaning of the rows of the design sensitivity matrix.

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N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA200A $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA200A $$$$$$$$ $ $ $ MODEL DESCRIPTION FULL SPAN 30 DEG FWD SWEPT WING $ $ WITH AILERON, CANARD AND AFT SWEPT $ $ VERTICAL FIN AND RUDDER. $ $ BAR MODEL WITH DUMBBELL MASSES. $ $ $ $ SOLUTION QUASI-STEADY AEROELASTIC ANALYSIS $ $ AND UNSTEADY FLUTTER ANALYSIS USING $ $ DOUBLET-LATTICE METHOD $ $ AERODYNAMICS AT MACH NO. 0.9. $ $ $ $ OUTPUT STANDARD AEROELASTIC OUTPUT PLUS $ $ A TABLE IDENTIFYING RESPONSES $ $ FOR WHICH SENSITIVITY RESULTS ARE $ $ AVAILABLE FOLLOWED BY A MATRIX OF $ $ SENSITIVITY VALUES. $ $ $ $ $ $$$$$$$$ $$$$$$$$ TIME 30 $ CPU TIME IN MINUTES SOL 200 $ OPTIMIZATION WITH AEROELASTICITY CEND EXAMPLE HA200A: 30 DEG FWD SWEPT WING WITH CANARD AND FIN PAGE 3 DEMONSTRATION OF AEROELASTIC SENSITIVITY ANALYSIS C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA200A: 30 DEG FWD SWEPT WING WITH CANARD AND FIN 2 SUBTI = DEMONSTRATION OF AEROELASTIC SENSITIVITY ANALYSIS 3 ECHO = BOTH 4 SPC = 1 5 DESOBJ = 10 6 DISP = ALL $ 7 STRESS = ALL $ 8 FORCE = ALL $ 9 AEROF = ALL $ 10 APRES = ALL $ 11 SUBCASE 1 12 LABEL = SUBSONIC SYMMETRIC PULLOUT 13 ANALYSIS = SAERO 14 DESSUB = 1 15 TRIM = 1 $ 16 SUBCASE 2 17 LABEL = SUPERSONIC SYMMETRIC PULLOUT 18 ANALYSIS = SAERO 19 DESSUB = 2 20 TRIM = 2 $ 21 SUBCASE 3 22 LABEL = HIGH SPEED ROLLING PULLOUT 23 ANALYSIS = SAERO 24 DESSUB = 200 25 TRIM = 3 $ 26 SUBCASE 4 27 LABEL = HIGH SPEED PULLUP WITH ABRUPT ROLL 28 ANALYSIS = SAERO 29 DESSUB = 200 30 TRIM = 4 $ 31 SUBCASE 5 32 LABEL = SUBSONIC ENTRY INTO SNAP ROLL 33 ANALYSIS = SAERO 34 DESSUB = 200 35 TRIM = 5 $ 36 SUBCASE 6 37 LABEL = SUBSONIC FLUTTER ANALYSIS 38 ANALYSIS = FLUTTER 39 SET 10 = 1,THRU,100000 40 PARAM OPPHIPA,1 41 DISP = 10 42 STRESS = NONE $ 43 FORCE = NONE $ 44 AEROF = NONE $ 45 APRES = NONE $ 46 DESSUB = 6 47 METHOD = 20 48 FMETHOD = 30 49 SUBCASE 7 50 LABEL = SUPERSONIC FLUTTER ANALYSIS 51 ANALYSIS = FLUTTER 52 DISP = NONE $ 53 STRESS = NONE $ 54 FORCE = NONE $ 55 AEROF = NONE $ 56 APRES = NONE $ 57 DESSUB = 7 58 METHOD = 20 59 FMETHOD = 40 60 BEGIN BULK

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EXAMPLE HA200A: 30 DEG FWD SWEPT WING WITH CANARD AND FIN PAGE 5 DEMONSTRATION OF AEROELASTIC SENSITIVITY ANALYSIS I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 $ INCLUDE FSW_TWO.DAT $ $ * RIGHT WING STRUCTURE * $ $ THE CBAR ENTRY DEFINES A SIMPLE BEAM ELEMENT. LISTED ARE $ ITS PROPERTY ENTRY ID, THE TWO GRID POINTS JOINED BY THE $ BEAM AND COMPONENTS OF A VECTOR FROM THE FIRST POINT. $ THIS VECTOR DEFINES THE DIRECTION OF THE STRUCTURAL DE$ FLECTION OF THE POINT AND ITS POSITIVE SENSE. $ $ EID PID GA GB X1,GO X2 X3 CBAR 110 101 100 110 0. 0. 1. CBAR 120 102 110 120 0. 0. 1. $ $ THE RBAR ENTRY DEFINES A RIGID BAR. LISTED ARE THE GRID $ POINTS AT EACH END AND THE DEPENDENT AND INDEPENDENT DOFS $ AT EACH END. THE NUMBER OF INDEPENDENT DOFS AT THE TWO $ ENDS MUST EQUAL SIX. BY DEFAULT THOSE NOT DECLARED INDE$ PENDENT ARE MADE DEPENDENT. $ $ EID GA GB CNA CNB CMA CMB RBAR 111 110 111 123456 RBAR 112 110 112 123456 RBAR 121 120 121 123456 RBAR 122 120 122 123456 $ $ THE PBAR ENTRY DEFINES GEOMETRIC PROPERTIES OF THE BEAM. $ LISTED ARE ITS ASSOCIATED MATERIAL ENTRY ID, ITS CROSS SEC$ TIONAL AREA, AREA MOMENTS OF INERTIA, TORSIONAL MOMENT $ OF INERTIA AND NON-STRUCTURAL MASS PER UNIT AREA. THE $ OPTIONAL CONTINUATION ENTRY CONTAINS STRESS RECOVERY $ COEFFICIENTS, I.E., Y,Z COORDINATES WHERE STRESSES ARE $ TO BE COMPUTED. K1 AND K2 ARE AREA FACTORS FOR SHEAR $ STIFFNESS (DEFAULT IS BLANK; THEN SHEAR STIFFNESS IS $ INFINITE, I.E., SHEAR FLEXIBILITY IS ZERO. I12 IS THE $ AREA PRODUCT OF INERTIA. $ $ INBOARD WING $ $ PID MID A I1 I2 J NSM PBAR 101 2 1.5 0.173611+2.0 0.462963 $ C1 C2 D1 D2 E1 E2 F1 F2 +PB1W 0.5 3.0 0.5 -3.0 -0.5 3.0 -0.5 -3.0 $ K1 K2 I12 +PB2W 0.0 $ $ OUTBOARD WING $ $ PID MID A I1 I2 J NSM PBAR 102 2 1.5 0.173611+2.0 0.462963 $ C1 C2 D1 D2 E1 E2 F1 F2 +PB3W 0.5 3.0 0.5 -3.0 -0.5 3.0 -0.5 -3.0 $ K1 K2 I12 +PB4W 0.0 $ $ * LEFT WING STRUCTURE * $ $ EID PID GA GB X1,GO X2 X3 CBAR 210 101 100 210 0. 0. 1. CBAR 220 102 210 220 0. 0. 1. $ $ EID GA GB CNA CNB CMA CMB RBAR 211 210 211 123456 RBAR 212 210 212 123456 RBAR 221 220 221 123456 RBAR 222 220 222 123456 $ $

* FIN STRUCTURE * $ $ CBAR $ $ PBAR $ +PB1V $ +PB2V $ $ RBAR RBAR $ $ $ $ $

.

$ $ $ $ $ $ $ $

$ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $ $ $

+PB1W +PB2W

+PB3W +PB4W $ $ $

$ $ $

EID 310

PID 103

GA 100

GB 310

X1,GO 0.

X2 0.

X3 1.

PID 103 C1 0.5 K1

MID 3 C2 3.0 K2

A 1.5 D1 0.5 I12 0.0

I1 I2 0.173611+2.0 D2 E1 -3.0 -0.5

J NSM 0.462963 E2 F1 3.0 -0.5

EID 311 312

GA 310 310

GB 311 312

CNA 123456 123456

CMA

$

CNB

+PB1V F2 -3.0

+PB2V

CMB

THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES. LISTED ARE ITS ID, ITS ELASTIC MODULUS, SHEAR MODULUS, POISSONS RATIO, MASS DENSITY, TEMPERATURE EXPANSION COEFFICIENT,

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$ REFERENCE TEMPERATURE AND A STRUCTURAL DAMPING COEFFICIENT. $ $ DENSITY HAS BEEN ADJUSTED TO GIVE A WEIGHT OF 333.333 LBS FOR THE $ INBOARD BAR AND 666.667 LBS FOR THE OUTBOARD BAR WHEN X1 = X2 = 1.0 $ AND 50.0 LBS FOR THE FIN WHEN X3 = 1.0 $ $ MID E G NU RHO A TREF GE MAT1 2 1.44+9 5.40+8 38.49002 $+MAT1 ST SC SS MCSID +MT2 7.20+6 5.76+6 MAT1 3 1.44+9 5.40+8 5.773503 +MT3 7.20+6 5.76+6 $ $ * WING AND FIN MASSES * $ $ THE CONM2 ENTRY DEFINES A CONCENTRATED MASS. LISTED ARE $ ITS ID, GRID LOCATION, COORDINATE SYSTEM TO LOCATE THE $ CENTER OF GRAVITY, THE MASS VALUE AND THE LOCATION OF $ THE CENTER OF GRAVITY RELATIVE TO THE GRID LOCATION. $ $ RIGHT WING $ CONM2 111 111 0 300.0 CONM2 112 112 0 200.0 CONM2 121 121 0 300.0 CONM2 122 122 0 200.0 $ $ LEFT WING $ CONM2 211 211 0 300.0 CONM2 212 212 0 200.0 CONM2 221 221 0 300.0 CONM2 222 222 0 200.0 $ $ FIN $ CONM2 311 311 0 30.0 CONM2 312 312 0 20.0 $ $

* * TRIM CONDITIONS * * $ $ $ $ $ $ $ $ $ $ $ $ TRIM $ +TR1A +TR1B $ $ $ TRIM +TR2A +TR2B $ $ $ TRIM +TR3A +TR3B $ $ $ TRIM +TR4A +TR4B $ $ $ TRIM +TR5A +TR5B $ $ $ $ $ $ $ $ $ $ $

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TRIM CONDITION 1: SUBSONIC SYMMETRIC PULLOUT MACH 0.9 UX3 0.0 0.0

Q 1200.0 ETC AILERON URDD6

LABEL1 PITCH

UX1 0.0

LABEL2 URDD3

UX2 -6.0

0.0 0.0

RUDDER

0.0

URDD2

0.0

1.2 0.0 0.0

863.0 PITCH AILERON 0.0 URDD6 0.0

0. RUDDER

URDD3 0.0

-4.0 URDD2

0.9 0. 0.

0.9 0. 0.

1200.0 PITCH 6.0499-4URDD3 AILERON .174533 YAW 0. URDD6 0.

1200.0 PITCH 6.0499-4URDD3 AILERON .174533 ROLL 0. URDD6 0.

300.0 SIDES RUDDER

$

$

$

ANGLEA .0031512ELEV 0. ROLL 0. .174533

$ $ $ $ $ $ $ $ $ $ $ +TRM +TR1A +TR1B

0.

$ $ $ +TR3A +TR3B

0.

$ $ $ +TR4A +TR4B

.174533 YAW 0.

$ $ $ +TR5A +TR5B

-4.8 URDD2

-4.8 YAW

TRIM CONDITION 5: SUBSONIC ENTRY INTO SNAP-ROLL 5 0.45 PITCH 0. AILERON 0.

$

+TR2A +TR2B

TRIM CONDITION 4: HIGH SPEED PULLUP WITH ABRUPT ROLL 4 URDD5 URDD2

$ $ $ $ $ $ $ $

0.0

TRIM CONDITION 3: HIGH SPEED ROLLING PULLOUT 3 URDD5 URDD4

+MT3

$ $

TRIM CONDITION 2: SUPERSONIC SYMMETRIC PULLOUT 2 URDD5 URDD4

+MT2

$

THE TRIM ENTRY SPECIFIES CONSTRAINTS FOR THE TRIM VARIABLES LISTED ON THE AESTAT AND AESURF ENTRYS. LISTED ARE ITS ID, THE MACH NUMBER, DYNAMIC PRESSURE AND PAIRS OF TRIM VARIABLES AND THEIR CONSTRAINED VALUES. THOSE THAT ARE NOT HELD FIXED MUST BE CONSTRAINED BY REACTION FORCES STIPULATED ON THE SUPORT ENTRY. SEE SECTION 3.5.3 OF THE THEORETICAL MANUAL FOR MORE DETAILS.

ID 1 LABEL3 URDD5 URDD4

$ $

$ $

* * *

$ * * *

FLUTTER ANALYSIS

* * *

* AERODYNAMICS * THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE REFERENCE LENGTHS PLUS SYMMETRY KEYS. SYMXZ = 0 INDICATES THAT THE MODEL IS MOUNTED WITH NO ROOT REFLECTION PLANE;

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$ $ $

Aeroelastic Design Sensitivities and Optimization

$ SYMXY = 0 INDICATES THAT THE MODEL IS MOUNTED FAR ENOUGH $ FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE NEGLIGIBLE. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 10.0 2.378-3 $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ TO GENERATE GENERALIZED AERO FORCE MATRICES. IF MORE THAN $ EIGHT MACH NOS OR REDUCED FREQUENCIES ARE REQUIRED A SECOND $ MKAERO1 IS NECESSARY. $ $MKAERO1 M1 M2 ETC MKAERO1 0.90 1.20 $+MK K1 K2 K3 K4 K5 ETC +MK 0.001 0.01 0.1 0.3 0.5 1.0 $ $ $ * VIBRATION ANALYSIS * $ $ THE EIGR ENTRY SPECIFIES THE METHOD OF EXTRACTING THE EIGEN$ SOLUTIONS OF THE STRUCTURE IN A VACUUM, IN THIS CASE AN $ AUTOMATIC SELECTION OF EITHER THE GIVENS METHOD OR THE $ MODIFIED GIVENS METHOD. THREE MODES ARE DESIRED, NORMAL$ IZED ON THE MAXIMUM DISPLACEMENTS. $ $EIGR SID METHOD F1 F2 NO EIGR 20 AGIV 15 $ NORM G C +AGIV MAX $ $ THE PARAM, OPPHIPA = 1 (WHEN ACCOMPANIED BY A DISPLACEMENT $ REQUEST IN THE CASE CONTROL SECTION) WILL GENERATE THE $ VIBRATION MODE DISPLACEMENTS FOR BOTH THE STRUCTURAL GRID $ POINTS AND THE AERODYNAMIC GRID POINTS (BOX CENTERLINE $ MIDPOINTS) IN THE OUTPUT. $ PARAM OPPHIPA 1 $ $ $ * * FLUTTER SOLUTION PARAMETERS * * $ $ THE FLUTTER ENTRY DEFINES THE METHOD OF SOLUTION, IDENTIFIES $ THE FLFACT ENTRIES THAT FOLL0W, SPECIFIES THE INTERPOLATION $ METHOD, THE NUMBER OF ROOTS DESIRED IN THE OUTPUT AND THE $ CRITERION FOR CONVERGENCE (DEFAULT IS 10-3). $ $ SID METHOD DENS MACH VEL IMETH NVALUE EPS FLUTTER 30 PK 1 2 3 S 8 $ $ FLFACT ENTRIES ARE USED TO SPECIFY DENSITY RATIOS, MACH NOS $ AND REDUCED FREQUENCIES/VELOCITIES FOR FLUTTER ANALYSES. $ NEGATIVE VELOCITIES ARE SIGNALS TO COMPUTE AND PRINT EIGEN$ VECTORS. $ $ SID F1 F2 F3 F4 F5 F6 F7 FLFACT 1 1.0 FLFACT 2 0.90 FLFACT 3 1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 FLUTTER 40 PK 11 12 13 S 8 FLFACT 11 1.0 FLFACT 12 1.20 FLFACT 13 1000.0 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 $ $ THE PARAM, LMODES, N, ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE FLUTTER ANALYSIS. $ PARAM LMODES 12 $ $ * * * * $ $ * * * OPTIMIZATION * * * $ $ $ * * * $ $ * THE DESIGN MODEL * $ $ DEFINITION OF THE DESIGN VARIABLES $ $ THE DESVAR ENTRY DEFINES A DESIGN VARIABLE FOR DESIGN $ OPTIMIZATION. LISTED ARE A UNIQUE DESIGN VARIABLE ID $ NUMBER, A USER SUPPLIED NAME FOR PRINTING PURPOSES, AN $ INITIAL VALUE, A LOWER BOUND, AND AN UPPER BOUND. $ $DESVAR ID LABEL XINIT XLB XUB DESVAR 10 PBAR101 1.0 0.001 100.0 DESVAR 20 PBAR102 1.0 0.001 100.0 DESVAR 30 PBAR103 0.1 0.001 100.0 $

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $

$ $ $ $ $ $ +MK +MK

$ $ $ $ $ $ $ $ +EIGR +AGIV $

$ $ $ $ $ $ $ $ $ $ $ $ $ $ DENSITY MACH VELOCITY

8-13

Chapter 8

Aeroelastic Design Sensitivities and Optimization

$ RELATIONSHIP OF DESIGN VARIABLES TO ANALYSIS MODEL PROPERTIES $ $ THE DVPREL1 ENTRY EXPRESSES AN AN ANALYSIS MODEL PROPERTY AS $ A LINEAR FUNCTION OF DESIGN VARIABLES. IT LISTS A UNIQUE ID, $ AN ANALYSIS MODEL ENTRY TYPE ID STRING, A PROPERTY ENTRY ID, $ THE FIELD POSITION OF THE PROPERTY ENTRY OR WORD POSITION IN $ THE ELEMENT PROPERTY TABLE OF THE ANALYSIS MODEL, THE MINIMUM $ AND MAXIMUM VALUES ALLOWED FOR THIS PROPERTY DURING OPTIMIZA TION, $ A CONSTANT TERM OF RELATION, A DESIGN VARIABLE ENTRY (DESVAR) $ ID, AND A COEFFICIENT OF LINEAR RELATION. THE EQUATION IS PI $ = CO + CJXJ AND THE CONTINUATION ENTRY CAN BE USED TO LIST $ MORE THAN ONE CJ. IN THIS CASE, THERE IS ONLY ONE CJ FOR EACH $ DVPREL1 ENTRY. $ $DVPREL1ID TYPE PID FID PMIN PMAX C0 XXXX +DVPREL1 DVPREL1 1014 PBAR 101 4 $ DVID1 COEF1 DVID2 COEF2 DVID3 COEF3 .... 10 1.5 DVPREL1 1015 PBAR 101 5 10 0.173611 DVPREL1 1016 PBAR 101 6 10 2.0 DVPREL1 1017 PBAR 101 7 10 0.462963 DVPREL1 1024 PBAR 102 4 20 1.5 DVPREL1 1025 PBAR 102 5 20 0.173611 DVPREL1 1026 PBAR 102 6 20 2.0 DVPREL1 1027 PBAR 102 7 20 .462963 DVPREL1 1028 PBAR 103 4 30 1.5 DVPREL1 1029 PBAR 103 5 30 0.173611 DVPREL1 1030 PBAR 103 6 30 2.0 DVPREL1 1031 PBAR 103 7 30 .462963 $ $ * * * $ $ * STRUCTURAL RESPONSES AND CONSTRAINTS * $ $ THE DRESP1 ENTRY DEFINES A SET OF STRUCTURAL RESPONSES THAT $ IS USED IN THE DESIGN EITHER AS CONSTRAINTS OR AS AN OBJEC TIVE. $ IT LISTS A UNIQUE ENTRY IDENTIFIER, A USER DEFINED LABEL, THE $ THE RESPONSE TYPE, THE PROPERTY ENTRY TYPE OR ELEMENT ID FLAG $ (ELEM), A REGION IDENTIFIER FOR CONSTRAINT SCREENING, AND A $ NUMBER OF ATTRIBUTES DEFINED IN THE TABLE GIVEN IN THE DRESP1 $ BULK DATA ENTRY DESCRIPTION (SEE THE DESIGN OPTIMIZATION USERS GUIDE $ OR THE QUICK REFERENCE GUIDE). $ $ $ DRESP1,10, IDENTIFIES THE $ WEIGHT RESPONSE USED AS THE OBJECTIVE FUNCTION. $ $DRESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 DRESP1 10 WEIGHT WEIGHT $ $ THE FOLLOWING DRESP1 ENTRIES IDENTIFY MAXIMUM AND MINIMUM $ STRESSES AT END A OF ALL ELEMENTS IN PROPERTY GROUPS 101, $ 102, AND 103. $ STRESS RESPONSES $ $DRESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 DRESP1 1001 TEN110 STRESS PBAR 7 101 102 103 DRESP1 1002 COM110 STRESS PBAR 8 101 102 103 $ $ STRESS CONSTRAINTS $ $ THE DCONSTR ENTRY DEFINES DESIGN CONSTRAINTS. LISTED ARE $ A CONSTRAINT SET ID, THE DRESPI ENTRY ID AND THE LOWER AND $ UPPER BOUND IMPOSED ON THIS RESPONSE QUANTITY. THE FOLLOWING $ TWO ENTRIES IMPOSE LIMITS ON THE ALLOWABLE STRESSES IN THE $ BARS IN UNITS OF POUNDS/FOOT**2. $ $DCONSTR DCID RID LALLOW UALLOW DCONSTR 200 1001 -5.76+6 7.20+6 DCONSTR 200 1002 -5.76+6 7.20+6 $ $ $ TIP TWIST RESPONSES $ DRESP1 101 RTIPROT DISP 5 120 DRESP1 201 LTIPROT DISP 5 220 DRESP1 100 RTROT DISP 5 100 $ $ SECOND LEVEL RESPONSES FOR TIP TWIST $

8-14

NX Nastran Aeroelastic Analysis User’s Guide

Aeroelastic Design Sensitivities and Optimization

$ THE DRESP2 ENTRY DEFINES THE INPUT ARGUMENTS TO USER-SUPPLIED $ EQUATIONS. THESE SECOND-LEVEL RESPONSES CAN THEN EITHER BE $ USED AS CONSTRAINTS OR AS AN OBJECTIVE FUNCTION. INPUT MAY $ CONSIST OF DESIGN VARIABLES (DESVAR), FIRST-LEVEL (DRESP1) $ RESPONSES, TABLE CONSTANTS (DTABLE), AND GRID COORDINATES $ (DVGRID). IT LISTS AN ID, A USER DEFINED LABEL, THE DEQATN $ ENTRY ID, A REGION IDENTIFIER FOR CONSTRAINT SCREENING, A $ STRING INDICATING DESVAR ID NUMBERS, A DESVAR ID, A STRING $ INDICATING THAT THE LABELS FOR THE CONSTANTS IN A DTABLE ENTRY $ FOLLOW, THE LABELS OF CONSTANTS IN THE DTABLE INPUT, $ A STRING INDICATING DRESP1 ID NUMBERS, DRESP1 IDS, A STRING $ SIGNIFYING THAT THE IDS AND DIRECTIONS FOLLOWING ARE NODE $ NUMBERS AND CARTESIAN DIRECTION COMPONENTS, NODE NUMBERS, $ AND CARTESIAN DIRECTIONS. $ $DRESP2 ID LABEL EQID REGION XXXX XXXX XXXX XXXX DRESP2 5 RHSTWIST5 +DR25 $ DRESP1 NR1 NR2 NR3 ETC +DR25 DRESP1 101 100 $ DRESP2 6 LHSTWIST5 +DR26 +DR26 DRESP1 201 100 $ $ EQUATION DEFINING SECOND LEVEL RESPONSE FOR TIP TWIST $ $ THE DEQATN ENTRY DEFINES THE EQUATION(S) USED IN THE DESIGN $ PROCESS. IT LISTS A UNIQUE EQUATION ID, AND THE EQUATION IS $ WRITTEN IN FORTRAN LIKE SYNTAX FOLLOWING THE RULES IN DEFIN ING $ DMAP ASSIGNMENTS AND FUNCTIONS. $ $DEQATN EQID EQUATION DEQATN 5 F(RTIP,RROOT) = RTIP - RROOT $ $ $ TIP TWIST CONSTRAINTS $ DCONSTR 50 5 -0.017450.01745 DCONSTR 50 6 -0.017450.01745 DCONSTR 60 5 -.008726 .008726 DCONSTR 60 6 -.008726 .008726 DCONADD 1 50 200 DCONADD 2 60 200 $ $ * * * $ $ * AILERON ROLL EFFECTIVENESS * $ $ $ RESPONSES REQUIRED FOR ROLL EFFECTIVENESS $ DRESP1 1401 CLDELTA STABDER 517 0 4 DRESP1 1402 CLP STABDER 513 0 4 $ $ $DRESP2 ID LABEL EQID REGION XXX XXX XXX XXX +DRES2 DRESP2 2401 ROLLEFF 103 +DR2401 $ DRESP1 NR1 NR2 NR3 ETC +DR2401 DRESP1 1402 1401 $DEQATN EQID EQUATION DEQATN 103 F(A,B) = -B/A $ $ SUBSONIC AILERON EFFECTIVENESS CONSTRAINT $ DCONSTR 50 2401 0.60 $ $ SUBSONIC AILERON EFFECTIVENESS CONSTRAINT $ DCONSTR 60 2401 0.43 $ $ * * * $ $ * SUBSONIC AND SUPERSONIC FLUTTER * $ $ RESPONSE FOR SUBSONIC FLUTTER $ $DRESP1 ID LABEL FLUTTER XXXX REGION SID XXXX ID_MODE +DR DRESP1 1 FLUTTER FLUTTER 88 +DR1 $ ID_DENS ID_MACH ID_VEL +DR1 1 2 4 $ $ SELECTION OF FLUTTER MODES FOR OPTIMIZATION $ SET1 88 4 THRU 7 $ $ SELECTION OF VELOCITIES FOR IMPOSING FLUTTER CONSTRAINTS $ FLFACT 4 1000.0 1300.0 1500.0 $ $ EQUATION FOR SECOND LEVEL FLUTTER RESPONSE TO CONSTRAIN SYSTEM DAMP ING $ DRESP2 4 GDAMP 4 +DR24

NX Nastran Aeroelastic Analysis User’s Guide

8-15

Chapter 8

Aeroelastic Design Sensitivities and Optimization

+DR24 DRESP1 1 DEQATN 4 F(A) = (A - 0.03)/0.1 $ $ CONSTRAINT ON AEROELASTIC SYSTEM DAMPING $ DCONSTR 6 4 -1.0+20 -0.3 $ $ $ RESPONSE FOR SUPERSONIC FLUTTER $ $DRESP1 ID LABEL FLUTTER XXXX REGION SID XXXX ID_MODE +DR DRESP1 11 FLUTTER FLUTTER 89 +DR11 $ ID_DENS ID_MACH ID_VEL +DR11 11 12 14 $ SET1 89 4 THRU 7 $ FLFACT 14 1000.0 1300.0 1500.0 $ DRESP2 21 GDAMP 4 +DR21 +DR21 DRESP1 11 $ DCONSTR 7 21 -1.0+20 -0.3 $ $ * * * $ $ * OPTIMIZATION CONTROL PARAMETERS * $ $ THE DOPTPRM ENTRY IS USED TO OVERRIDE DEFAULT OPTIMIZATION $ PARAMETERS. THE DOPTPRM ENTRY IS USED $ HERE TO SET THE MAXIMUM NUMBER OF DESIGN CYCLES TO 25 AND TO $ REQUEST DETAILED RESULTS FOLLOWING EACH OPTIMIZATION CYCLE. $ $ PARAM1 VAL1 PARAM2 VAL2 ETC DOPTPRM DESMAX 25 P1 2 P2 15 DELB 0.01 $ $ THE PARAMETER CDIF,YES FORCES THE SELECTION OF THE CENTRAL $ DIFFERENCE SCHEME USED IN THE SEMI-ANALYTIC APPROACH $ REGARDLESS OF THE TYPE OF OPTIMIZATION REQUESTED. PARAM CDIF YES $ $ THE BULK DATA PARAMETER, NASPRT, IS USED TO SPECIFY HOW OFTEN $ NX NASTRAN OUTPUT IS TO BE PROVIDED. ITS VALUE INDICATES THAT $ OUTPUT IS TO BE COMPUTED EVERY N-TH DESIGN CYCLE. PARAM NASPRT 2 $ $ THE BULK DATA PARAMETER, OPTEXIT, IS USED TO CONTROL THE EXIT $ POINT FROM A SOLUTION 200 OPTIMIZATION TASK. OPTEXIT=4 INDI CATES $ THAT A SENSITIVITY ANALYSIS IS TO BE PERFORMED WITHOUT OPTI MIZATION. PARAM OPTEXIT 4 $ $ THE DSCREEN ENTRY IS USED HERE TO FORCE RETENTION OF ALL THE $ CONSTRAINTS SO THAT THE DESIRED RESPONSE SENSITIVITIES WILL $ BE PRINTED. CONSTRAINTS ARE RETAINED IF THEY ARE GREATER $ THAN TRS. $SCREEN RTYPE TRS NSTR DSCREEN STRESS -2.0 DSCREEN EQUA -2.0 ENDDATA INPUT BULK DATA CARD COUNT = 900

8-16

NX Nastran Aeroelastic Analysis User’s Guide

Aeroelastic Design Sensitivities and Optimization

Listing 8-1. Input File for Design Sensitivities of FSW Airplane EXAMPLE HA200A: 30 DEG FWD SWEPT WING WITH CANARD AND FIN PAGE 23 DEMONSTRATION OF AEROELASTIC SENSITIVITY ANALYSIS S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 1AELIST 1000 1000 THRU 1007 2AELIST 1100 1119 1123 1127 1131 3AELIST 2000 2000 THRU 2007 4AELIST 2100 2103 2107 2111 2115 5AELIST 3000 3103 3107 3111 3115 6AERO 1 10.0 2.378-3 7AEROS 1 100 10.0 40.0 400.0 8AESTAT 501 ANGLEA 9AESTAT 502 PITCH 10AESTAT 503 URDD3 11AESTAT 504 URDD5 12AESTAT 511 SIDES 13AESTAT 512 YAW 14AESTAT 513 ROLL 15AESTAT 514 URDD2 16AESTAT 515 URDD4 17AESTAT 516 URDD6 18AESURF 505 ELEV 1 1000 1 2000 19AESURF 517 AILERON 110 1100 210 2100 20AESURF 518 RUDDER 301 3000 21CAERO1 1000 1000 2 4 1 22+CARC 10. 0. 0. 10. 10. 5. 0. 10. 23CAERO1 1100 1000 8 4 1 24+CARW 25. 0. 0. 10. 13.45299+20. 0. 10. 25CAERO1 2000 1000 2 4 1 26+CALC 10. -5. 0. 10. 10. 0. 0. 10. 27CAERO1 2100 1000 8 4 1 28+CALW 13.45299-20. 0. 10. 25. 0. 0. 10. 29CAERO1 3100 1000 4 4 1 30+CA1FI 30.7735 0. 10. 10. 25. 0. 0. 10. 31CBAR 101 100 97 98 0. 0. 1. 32CBAR 102 100 98 90 0. 0. 1. 33CBAR 103 100 90 99 0. 0. 1. 34CBAR 104 100 99 100 0. 0. 1. 35CBAR 110 101 100 110 0. 0. 1. 36CBAR 120 102 110 120 0. 0. 1. 37CBAR 210 101 100 210 0. 0. 1. 38CBAR 220 102 210 220 0. 0. 1. 39CBAR 310 103 100 310 0. 0. 1. 40CONM2 97 97 0 3000.0 41CONM2 98 98 0 3000.0 42CONM2 99 99 0 3000.0 43CONM2 100 100 0 3000.0 44CONM2 111 111 0 300.0 45CONM2 112 112 0 200.0 46CONM2 121 121 0 300.0 47CONM2 122 122 0 200.0 48CONM2 211 211 0 300.0 49CONM2 212 212 0 200.0 50CONM2 221 221 0 300.0 51CONM2 222 222 0 200.0 52CONM2 311 311 0 30.0 53CONM2 312 312 0 20.0 54CORD2R 1 0 12.5 0. 0. 12.5 0. 10. 55+CRD1 20. 0. 0. 56CORD2R 2 0 30. 0. 0. 30. 0. 10. 57+CRD2RW 38.66025+5.0 0. 58CORD2R 20 0 30. 0. 0. 30. 0. 10. 59+CRD2LW 38.66025-5.0 0. 60CORD2R 100 0 15.0 0.0 0.0 15.0 0.0 -10.0 61+CRD100 0.0 0.0 0.0 62CORD2R 110 0 26.7265 10.0 0. 26.7265 10.0 -10.0 63+CRD2RA 36.7265 15.7735 0. 64CORD2R 210 0 26.7265 -10.0 0. 26.7265 -10.0 10.0 65+CRD2LA 36.7265 -15.77350. 66CORD2R 300 0 30.0 0. 0. 30.0 10.0 0. 67+CRD2FI 20.0 0.0 5.7735 68CORD2R 301 0 32.5 0. 0. 32.5 -10. 0.0 69+CRD2R 22.5 0. 5.7735 70DCONADD 1 50 200 71DCONADD 2 60 200 72DCONSTR 6 4 -1.0+20 -0.3 73DCONSTR 7 21 -1.0+20 -0.3 74DCONSTR 50 5 -0.017450.01745 75DCONSTR 50 6 -0.017450.01745 76DCONSTR 50 2401 0.60 77DCONSTR 60 2401 0.43 78DCONSTR 60 6 -.008726.008726 79DCONSTR 60 5 -.008726.008726 80DCONSTR 200 1002 -5.76+6 7.20+6 81DCONSTR 200 1001 -5.76+6 7.20+6 82DEQATN 4 F(A) = (A - 0.03)/0.1 83DEQATN 5 F(RTIP,RROOT) = RTIP - RROOT 84DEQATN 103 F(A,B) =-B/A 85DESVAR 10 PBAR101 1.0 0.001 100.0 86DESVAR 20 PBAR102 1.0 0.001 100.0 Nastran Aeroelastic Analysis User’s Guide NX 87DESVAR 30 PBAR103 0.1 0.001 100.0 88DOPTPRM DESMAX 25 P1 2 P2 15 DELB 0.01 89DRESP1 1 FLUTTER FLUTTER 88

..

10

+CARC +CARW +CALC +CALW +CA1FI

+CRD1 +CRD2RW +CRD2LW +CRD100 +CRD2RA +CRD2LA +CRD2FI +CRD2R

8-17 +DR1

.

Chapter 8

Aeroelastic Design Sensitivities and Optimization

9293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182-

8-18

DRESP1 11 +DR11 11 DRESP1 100 DRESP1 101 DRESP1 201 DRESP1 1001 ++000025102 DRESP1 1002 ++000027102 DRESP1 1401 DRESP1 1402 DRESP2 4 +DR24 DRESP1 DRESP2 5 +DR25 DRESP1 DRESP2 6 +DR26 DRESP1 DRESP2 21 +DR21 DRESP1 DRESP2 2401 +DR2401 DRESP1 DSCREEN EQUA DSCREEN STRESS DVPREL1 1014 ++00000110 DVPREL1 1015 ++00000310 DVPREL1 1016 ++00000510 DVPREL1 1017 ++00000710 DVPREL1 1024 ++00000920 DVPREL1 1025 ++00001120 DVPREL1 1026 ++00001320 DVPREL1 1027 ++00001520 DVPREL1 1028 ++00001730 DVPREL1 1029 ++00001930 DVPREL1 1030 ++00002130 DVPREL1 1031 ++00002330 EIGR 20 +AGIV MAX FLFACT 1 FLFACT 2 FLFACT 3 FLFACT 4 FLFACT 11 FLFACT 12 FLFACT 13 FLFACT 14 FLUTTER 30 FLUTTER 40 GRID 90 GRID 97 GRID 98 GRID 99 GRID 100 GRID 110 GRID 111 GRID 112 GRID 120 GRID 121 GRID 122 GRID 210 GRID 211 GRID 212 GRID 220 GRID 221 GRID 222 GRID 310 GRID 311 GRID 312 MAT1 1 MAT1 2 +MT2 7.20+6 MAT1 3 +MT3 7.20+6 MKAERO1 0.90 +MK 0.001 OMIT1 4 PAERO1 1000 PARAM AUNITS PARAM CDIF PARAM GRDPNT

NX Nastran Aeroelastic Analysis User’s Guide

FLUTTER FLUTTER 12 14 RTROT DISP RTIPROT DISP LTIPROT DISP TEN110 STRESS 103 COM110 STRESS 103 CLDELTA STABDER CLP STABDER GDAMP 4 1 RHSTWIST5 101 100 LHSTWIST5 201 100 GDAMP 4 11 ROLLEFF 103 1402 1401 -2.0 -2.0 PBAR 101 1.5 PBAR 101 0.173611 PBAR 101 2.0 PBAR 101 0.462963 PBAR 102 1.5 PBAR 102 0.173611 PBAR 102 2.0 PBAR 102 .462963 PBAR 103 1.5 PBAR 103 0.173611 PBAR 103 2.0 PBAR 103 .462963 AGIV 1.0 0.90 1000.0 1000.0 1.0 1.20 1000.0 1000.0 PK PK

1.44+9 1.44+9 5.76+6 1.44+9 5.76+6 1.20 0.01 110 .031081 YES 90

1100.0 1300.0

89 5 5 5 7

PBAR PBAR

100 120 220 101

8 517 513

101 0 0

+DR11

+000025 +000026 +000027 +000028

4 4 +DR24 +DR25 +DR26 +DR21 +DR2401

4

+000001 +000002 +000003 +000004 +000005 +000006 +000007 +000008 +000009 +000010 +000011 +000012 +000013 +000014 +000015 +000016 +000017 +000018 +000019 +000020 +000021 +000022 +000023 +000024 +AGIV

5 6 7 4 5 6 7 4 5 6 7 15

1200.0 1500.0

1300.0

1400.0

1500.0

1600.0

1100.0 1200.0 1300.0 1500.0 1 2 11 12 15. 0. 0. 0. 10. 0. 20. 0. 30. 0. 27.11325+5. 24.61325+5. 29.61325+5. 21.33975+15. 18.83975+15. 23.83975+15. 27.11325-5. 24.61325-5. 29.61325-5. 21.33975-15. 18.83975-15. 23.83975-15. 32.88675+0. 30.38675+0. 35.38675+0. 5.40+8 5.40+8

1300.0

1400.0

1500.0

1600.0

5.40+8

5.773503

3 S 13 S 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5. 5. 5. 0.0 38.49002

DENSITY MACH VELOCITY

8 8

+MT2 +MT3 +MK

0.1 120

0.3 210

0.5 220

1.0 310

Aeroelastic Design Sensitivities and Optimization

183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242-

PARAM PARAM PARAM PARAM PARAM PBAR +PB1F +PB2F PBAR +PB1W +PB2W PBAR +PB3W +PB4W PBAR +PB1V +PB2V RBAR RBAR RBAR RBAR RBAR RBAR RBAR RBAR RBAR RBAR SET1 SET1 SET1 SET1 SET1 SET1 SPC1 SPLINE2 +SPRC SPLINE2 +SPRW SPLINE2 +SPLC SPLINE2 +SPLW SPLINE2 +SP2FI SUPORT TRIM +TR1A +TR1B TRIM +TR2A +TR2B TRIM +TR3A +TR3B TRIM +TR4A +TR4B TRIM +TR5A +TR5B ENDDATA TOTAL COUNT=

LMODES NASPRT OPPHIPA OPTEXIT WTMASS 100 1.0

12 2 1 4 .031081 1 4.0 1.0 1.0 0.0 101 2 1.5 0.5 3.0 0.5 0.0 102 2 1.5 0.5 3.0 0.5 0.0 103 3 1.5 0.5 3.0 0.5 0.0 111 110 111 112 110 112 121 120 121 122 120 122 211 210 211 212 210 212 221 220 221 222 220 222 311 310 311 312 310 312 88 4 THRU 89 4 THRU 1000 98 99 1100 99 100 2100 99 100 3100 99 100 1 1 90 1501 1000 1000 1. -1. 1601 1100 1100 -1. -1. 2501 2000 2000 1. -1. 2601 2100 2100 -1. -1. 3100 3100 3100 -1. -1. 90 23456 1 0.9 1200.0 URDD5 0.0 AILERON URDD4 0.0 URDD6 2 1.2 863.0 URDD5 0.0 AILERON URDD4 0.0 URDD6 3 0.9 1200.0 URDD5 0. AILERON URDD4 0. URDD6 4 0.9 1200.0 URDD5 0. AILERON URDD2 0. URDD6 5 0.45 300.0 PITCH 0. SIDES AILERON 0. RUDDER

.347222 .30 -1.0 -1.0

1.0 1.0

-1.0

-1.0

+PB1F +PB2F

0.173611+2.0 -3.0 -0.5

0.462963 3.0 -0.5

-3.0

+PB1W +PB2W

0.173611+2.0 -3.0 -0.5

0.462963 3.0 -0.5

-3.0

+PB3W +PB4W

0.173611+2.0 -3.0 -0.5

0.462963 3.0 -0.5

-3.0

+PB1V +PB2V

123456 123456 123456 123456 123456 123456 123456 123456 123456 123456 7 7 111 211 311

112 212 312

121 221

122 222

1007

1000

0.

1.

1

+SPRC

1131

1100

0.

1.

2

+SPRW

2007

1000

0.

1.

1

+SPLC

2131

2100

0.

1.

20

+SPLW

3115

3100

0.

1.

300

+SP2FI

PITCH 0.0 0.0 PITCH 0.0 0.0 PITCH .174533 0. PITCH .174533 0. ANGLEA 0. .174533

0.0 RUDDER

URDD3 0.0

-6.0 URDD2

0.0

+TR1A +TR1B

0. RUDDER

URDD3 0.0

-4.0 URDD2

0.0

+TR2A +TR2B

6.0499-4URDD3 YAW 0.

-4.8 URDD2

0.

+TR3A +TR3B

6.0499-4URDD3 ROLL 0.

-4.8 YAW

0.

+TR4A +TR4B

.0031512ELEV ROLL 0.

.174533 YAW 0.

+TR5A +TR5B

243

NX Nastran Aeroelastic Analysis User’s Guide

8-19

Chapter 8

Aeroelastic Design Sensitivities and Optimization

Listing 8-2. Sorted Bulk Data for Design Sensitivities of FSW Airplane EXAMPLE HA200A: 30 DEG FWD SWEPT WING WITH CANARD AND FIN PAGE 31 DEMONSTRATION OF AEROELASTIC SENSITIVITY ANALYSIS ----COMPARISON BETWEEN INPUT PROPERTY VALUES FROM ANALYSIS AND DESIGN MODELS -----------------------------------------------------------------------------------------------------------PROPERTY PROPERTY FIELD ANALYSIS DESIGN LOWER UPPER DIFFERENCE TYPE ID ID VALUE VALUE BOUND BOUND FLAG --------------------------------------------------------------------------------------------------------PBAR 101 4 1.500000E+00 1.500000E+00 1.000000E-03 1.000000E+20 NONE PBAR 101 5 1.736110E-01 1.736110E-01 1.000000E-03 1.000000E+20 NONE PBAR 101 6 2.000000E+00 2.000000E+00 1.000000E-03 1.000000E+20 NONE PBAR 101 7 4.629630E-01 4.629630E-01 1.000000E-03 1.000000E+20 NONE PBAR 102 4 1.500000E+00 1.500000E+00 1.000000E-03 1.000000E+20 NONE PBAR 102 5 1.736110E-01 1.736110E-01 1.000000E-03 1.000000E+20 NONE PBAR 102 6 2.000000E+00 2.000000E+00 1.000000E-03 1.000000E+20 NONE PBAR 102 7 4.629630E-01 4.629630E-01 1.000000E-03 1.000000E+20 NONE PBAR 103 4 1.500000E+00 1.500000E-01 1.000000E-03 1.000000E+20 WARNING PBAR 103 5 1.736110E-01 1.736110E-02 1.000000E-03 1.000000E+20 WARNING PBAR 103 6 2.000000E+00 2.000000E-01 1.000000E-03 1.000000E+20 WARNING PBAR 103 7 4.629630E-01 4.629630E-02 1.000000E-03 1.000000E+20 WARNING 1. IF FIELD ID IS LESS THAN ZERO, IT IDENTIFIES THE WORD POSITION OF AN ENTRY IN EPT. 2. IF FIELD ID IS GREATER THAN ZERO, IT IDENTIFIES THE FIELD POSITION ON A PROPERTY BULK DATA ENTRY. 3. THE DIFFERENCE FLAG IS USED TO CHARACTERIZE DIFFERENCES BETWEEN ANALYSIS AND DESIGN MODEL PROPERTIES: IF THE FLAG IS NONE, THEN THERE IS NO SIGNIFICANT DIFFERENCE BETWEEN THE TWO VALUES. IF THE FLAG IS WARNING, THEN THE USER IS ADVISED THAT DIFFERENCES EXIST. IF THE FLAG IS FATAL, THEN THE DIFFERENCES ARE GREATER THAN 1.00000E+35 AND THE RUN WILL BE TERMINATED. SUBSONIC FLUTTER ANALYSIS SUBCASE 6 FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0000 9.9999996E+24 1.0000000E+03 -1.1658986E-07 0.0000000E+00 -8.0813088E-06 0.0000000E+00 0.0000 9.9999996E+24 1.1000000E+03 -1.1734011E-07 0.0000000E+00 -8.9466430E-06 0.0000000E+00 0.0000 9.9999996E+24 1.2000000E+03 -1.1816442E-07 0.0000000E+00 -9.8285373E-06 0.0000000E+00 0.0000 9.9999996E+24 1.3000000E+03 -1.1906335E-07 0.0000000E+00 -1.0728584E-05 0.0000000E+00 0.0000 9.9999996E+24 1.4000000E+03 -1.2003827E-07 0.0000000E+00 -1.1648464E-05 0.0000000E+00 0.0000 9.9999996E+24 1.5000000E+03 -1.2108912E-07 0.0000000E+00 -1.2589756E-05 0.0000000E+00 0.0000 9.9999996E+24 1.6000000E+03 -1.2221761E-07 0.0000000E+00 -1.3554227E-05 0.0000000E+00 POINT = 2 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0000 9.9999996E+24 1.0000000E+03 1.1699186E-05 0.0000000E+00 8.1091729E-04 0.0000000E+00 0.0000 9.9999996E+24 1.1000000E+03 1.1702067E-05 0.0000000E+00 8.9222880E-04 0.0000000E+00 0.0000 9.9999996E+24 1.2000000E+03 1.1704831E-05 0.0000000E+00 9.7357039E-04 0.0000000E+00 0.0000 9.9999996E+24 1.3000000E+03 1.1708045E-05 0.0000000E+00 1.0549909E-03 0.0000000E+00 0.0000 9.9999996E+24 1.4000000E+03 1.1713160E-05 0.0000000E+00 1.1366403E-03 0.0000000E+00 0.0000 9.9999996E+24 1.5000000E+03 1.1716822E-05 0.0000000E+00 1.2182096E-03 0.0000000E+00 0.0000 9.9999996E+24 1.6000000E+03 1.1721712E-05 0.0000000E+00 1.2999660E-03 0.0000000E+00 POINT = 3 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0001 1.5615369E+04 1.0000000E+03 5.6847133E-04 2.0384397E-03 3.6404604E-06 1.2807894E-02 0.0001 1.5798788E+04 1.1000000E+03 6.3057989E-04 2.2162513E-03 4.3904497E-06 1.3925118E-02 0.0001 1.6004385E+04 1.2000000E+03 5.6716142E-04 2.3866699E-03 4.2525448E-06 1.4995890E-02 0.0001 1.6233746E+04 1.3000000E+03 5.1665999E-04 2.5490285E-03 4.1374183E-06 1.6016020E-02 0.0001 1.6488756E+04 1.4000000E+03 5.4588803E-04 2.7026527E-03 4.6349355E-06 1.6981268E-02 0.0001 1.6771660E+04 1.5000000E+03 4.8446262E-04 2.8468547E-03 4.3328682E-06 1.7887317E-02 0.0001 1.7085139E+04 1.6000000E+03 4.2831604E-04 2.9809286E-03 4.0111213E-06 1.8729728E-02 POINT = 4 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0378 2.6484198E+01 1.0000000E+03 -2.6558921E-01 1.2018861E+00 -1.0028214E+00 7.5516729E+00 0.0378 2.6483675E+01 1.1000000E+03 -2.6536602E-01 1.3221009E+00 -1.1021984E+00 8.3070049E+00 0.0378 2.6483141E+01 1.2000000E+03 -2.6512387E-01 1.4423207E+00 -1.2013251E+00 9.0623684E+00 0.0378 2.6482637E+01 1.3000000E+03 -2.6486406E-01 1.5625439E+00 -1.3001850E+00 9.8177528E+00 0.0378 2.6482168E+01 1.4000000E+03 -2.6458707E-01 1.6827694E+00 -1.3987597E+00 1.0573153E+01 0.0378 2.6481766E+01 1.5000000E+03 -2.6429400E-01 1.8029947E+00 -1.4970340E+00 1.1328550E+01 0.0378 2.6481449E+01 1.6000000E+03 -2.6398593E-01 1.9232173E+00 -1.5949939E+00 1.2083931E+01 POINT = 5 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0746 1.3409112E+01 1.0000000E+03 -6.5904009E-01 2.3738327E+00 -4.9148674E+00 1.4915231E+01 0.0814 1.2283610E+01 1.1000000E+03 -6.3653028E-01 2.8504720E+00 -5.7001424E+00 1.7910044E+01 0.0926 1.0796389E+01 1.2000000E+03 -5.8971238E-01 3.5379593E+00 -6.5545511E+00 2.2229654E+01 0.1186 8.4333210E+00 1.3000000E+03 -3.3484232E-01 4.9067597E+00 -5.1616082E+00 3.0830082E+01 0.1069 9.3564539E+00 1.4000000E+03 -1.5777969E+00 4.7628493E+00 -2.3608471E+01 2.9925867E+01 0.0861 1.1615962E+01 1.5000000E+03 -2.3745260E+00 4.1104198E+00 -3.0662884E+01 2.5826530E+01 0.0623 1.6040276E+01 1.6000000E+03 -3.6939597E+00 3.1751065E+00 -3.6846848E+01 1.9949783E+01 POINT = 6 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.2559 3.9084766E+00 1.0000000E+03 -1.3052110E-02 8.1440907E+00 -3.3394367E-01 5.1170834E+01 0.2327 4.2964892E+00 1.1000000E+03 -1.4401533E-02 8.1494646E+00 -3.6871237E-01 5.1204601E+01 0.2013 4.9675179E+00 1.2000000E+03 -3.5838893E-01 7.6893902E+00 -8.6575775E+00 4.8313866E+01 0.1396 7.1633563E+00 1.3000000E+03 -6.4710283E-01 5.7766619E+00 -1.1743569E+01 3.6295837E+01 0.1177 8.4986610E+00 1.4000000E+03 1.6736831E-01 5.2435765E+00 2.7570887E+00 3.2946365E+01 0.1103 9.0680876E+00 1.5000000E+03 4.3815657E-01 5.2653308E+00 7.2477779E+00 3.3083050E+01 0.1030 9.7113571E+00 1.6000000E+03 6.4964896E-01 5.2443318E+00 1.0703327E+01 3.2951111E+01 POINT = 7 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.2850 3.5088842E+00 1.0000000E+03 -2.0820384E-01 9.0715408E+00 -5.9336195E+00 5.6998177E+01 0.2432 4.1113272E+00 1.1000000E+03 -2.6946324E-01 8.5164928E+00 -7.2095838E+00 5.3510704E+01 0.2135 4.6836166E+00 1.2000000E+03 -1.5785303E-02 8.1554890E+00 -4.0443879E-01 5.1242451E+01 0.1972 5.0697670E+00 1.3000000E+03 -1.7209273E-02 8.1621666E+00 -4.4128370E-01 5.1284409E+01 0.1833 5.4548392E+00 1.4000000E+03 -1.8678803E-02 8.1695127E+00 -4.7939676E-01 5.1330566E+01 0.1713 5.8387322E+00 1.5000000E+03 -2.0199422E-02 8.1775427E+00 -5.1893342E-01 5.1381016E+01 0.1607 6.2213368E+00 1.6000000E+03 -2.1776900E-02 8.1862755E+00 -5.6005710E-01 5.1435890E+01 POINT = 8 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK NX Nastran Aeroelastic Analysis User’s Guide 8-20 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.5795 1.7257444E+00 1.0000000E+03 -3.1901423E-02 1.8444788E+01 -1.8485602E+00 1.1589202E+02 0.5272 1.8967527E+00 1.1000000E+03 -3.4652915E-02 1.8460016E+01 -2.0096562E+00 1.1598771E+02

Aeroelastic Design Sensitivities and Optimization

0.3882 2.5761364E+00 1.5000000E+03 -4.4816069E-02 1.8534142E+01 -2.6094930E+00 1.1645345E+02 0.3643 2.7449362E+00 1.6000000E+03 -4.7589425E-02 1.8554012E+01 -2.7739472E+00 1.1657830E+02 SUPERSONIC FLUTTER ANALYSIS SUBCASE 7 ----IDENTIFICATION OF COLUMNS IN THE DESIGN SENSITIVITY --------MATRIX THAT ARE ASSOCIATED WITH DRESP1 ENTRIES --------- WEIGHT/VOLUME RESPONSES ---------------------------------------------COLUMN DRESP1 RESPONSE NO. ENTRY ID TYPE -----------------------------------------1 10 WEIGHT ----- STATICS RESPONSES -----------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE GRID/ELM COMPONENT SUB PLY NO. ENTRY ID TYPE ID NO. CASE NO. -------------------------------------------------------------------------------------------------2 100 DISP 100 5 1 3 101 DISP 120 5 1 4 201 DISP 220 5 1 5 1001 STRESS 110 7 1 6 1001 STRESS 210 7 1 7 1001 STRESS 120 7 1 8 1001 STRESS 220 7 1 9 1001 STRESS 310 7 1 10 1002 STRESS 110 8 1 11 1002 STRESS 210 8 1 12 1002 STRESS 120 8 1 13 1002 STRESS 220 8 1 14 1002 STRESS 310 8 1 ----- AEROELASTIC STABILITY DERIVATIVE RESPONSES ----------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE SUBCASE RESTRAINED/ COMPONENT AESTAT/AESURF NO. ENTRY ID TYPE ID UNRESTRAINED NO. ID ------------------------------------------------------------------------------------------------15 1401 STABDER 1 UNRESTRAINED 4 517 16 1402 STABDER 1 UNRESTRAINED 4 513 ----- STATICS RESPONSES -----------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE GRID/ELM COMPONENT SUB PLY NO. ENTRY ID TYPE ID NO. CASE NO. -------------------------------------------------------------------------------------------------17 100 DISP 100 5 2 18 101 DISP 120 5 2 19 201 DISP 220 5 2 20 1001 STRESS 110 7 2 21 1001 STRESS 210 7 2 22 1001 STRESS 120 7 2 23 1001 STRESS 220 7 2 24 1001 STRESS 310 7 2 25 1002 STRESS 110 8 2 26 1002 STRESS 210 8 2 27 1002 STRESS 120 8 2 28 1002 STRESS 220 8 2 29 1002 STRESS 310 8 2 ----- AEROELASTIC STABILITY DERIVATIVE RESPONSES -------------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE SUBCASE RESTRAINED/ COMPONENT AESTAT/AESURF NO. ENTRY ID TYPE ID UNRESTRAINED NO. ID -------------------------------------------------------------------------------------------------30 1401 STABDER 2 UNRESTRAINED 4 517 31 1402 STABDER 2 UNRESTRAINED 4 513 ----- STATICS RESPONSES ---------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE GRID/ELM COMPONENT SUB PLY NO. ENTRY ID TYPE ID NO. CASE NO. -------------------------------------------------------------------------------------------------32 1001 STRESS 110 7 3 33 1001 STRESS 210 7 3 34 1001 STRESS 120 7 3 35 1001 STRESS 220 7 3 36 1001 STRESS 310 7 3 37 1002 STRESS 110 8 3 38 1002 STRESS 210 8 3 39 1002 STRESS 120 8 3 40 1002 STRESS 220 8 3 41 1002 STRESS 310 8 3 42 1001 STRESS 110 7 4 43 1001 STRESS 210 7 4 44 1001 STRESS 120 7 4 45 1001 STRESS 220 7 4 ----IDENTIFICATION OF COLUMNS IN THE DESIGN SENSITIVITY --------MATRIX THAT ARE ASSOCIATED WITH DRESP1 ENTRIES --------- STATICS RESPONSES -------------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE GRID/ELM COMPONENT SUB PLY NO. ENTRY ID TYPE ID NO. CASE NO. ---------------------------------------------------------------------------------------------------46 1001 STRESS 310 7 4 47 1002 STRESS 110 8 4 48 1002 STRESS 210 8 4 49 1002 STRESS 120 8 4

NX Nastran Aeroelastic Analysis User’s Guide

8-21

Chapter 8

Aeroelastic Design Sensitivities and Optimization

50 1002 STRESS 220 8 4 51 1002 STRESS 310 8 4 52 1001 STRESS 110 7 5 53 1001 STRESS 210 7 5 54 1001 STRESS 120 7 5 55 1001 STRESS 220 7 5 56 1001 STRESS 310 7 5 57 1002 STRESS 110 8 5 58 1002 STRESS 210 8 5 59 1002 STRESS 120 8 5 60 1002 STRESS 220 8 5 61 1002 STRESS 310 8 5 ----- AEROELASTIC FLUTTER RESPONSES ------------------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE SUBCASE MODE DENSITY MACH VELOCITY NO. ENTRY ID TYPE ID NO. NO. --------------------------------------------------------------------------------------------------------62 1 FLUTTER 6 4 1.000 0.900 1000.000 63 1 FLUTTER 6 4 1.000 0.900 1300.000 64 1 FLUTTER 6 4 1.000 0.900 1500.000 65 1 FLUTTER 6 5 1.000 0.900 1000.000 66 1 FLUTTER 6 5 1.000 0.900 1300.000 67 1 FLUTTER 6 5 1.000 0.900 1500.000 68 1 FLUTTER 6 6 1.000 0.900 1000.000 69 1 FLUTTER 6 6 1.000 0.900 1300.000 70 1 FLUTTER 6 6 1.000 0.900 1500.000 71 1 FLUTTER 6 7 1.000 0.900 1000.000 72 1 FLUTTER 6 7 1.000 0.900 1300.000 73 1 FLUTTER 6 7 1.000 0.900 1500.000 74 11 FLUTTER 7 4 1.000 1.200 1000.000 75 11 FLUTTER 7 4 1.000 1.200 1300.000 76 11 FLUTTER 7 4 1.000 1.200 1500.000 77 11 FLUTTER 7 5 1.000 1.200 1000.000 78 11 FLUTTER 7 5 1.000 1.200 1300.000 79 11 FLUTTER 7 5 1.000 1.200 1500.000 80 11 FLUTTER 7 6 1.000 1.200 1000.000 81 11 FLUTTER 7 6 1.000 1.200 1300.000 82 11 FLUTTER 7 6 1.000 1.200 1500.000 83 11 FLUTTER 7 7 1.000 1.200 1000.000 84 11 FLUTTER 7 7 1.000 1.200 1300.000 ----IDENTIFICATION OF COLUMNS IN THE DESIGN SENSITIVITY --------MATRIX THAT ARE ASSOCIATED WITH DRESP1 ENTRIES --------- AEROELASTIC FLUTTER RESPONSES -------------------------------------------------------------------------------------------------------------COLUMN DRESP1 RESPONSE SUBCASE MODE DENSITY MACH VELOCITY NO. ENTRY ID TYPE ID NO. NO. ----------------------------------------------------------------------------------------------------------85 11 FLUTTER 7 7 1.000 1.200 1500.000 ----IDENTIFICATION OF COLUMNS IN THE DESIGN SENSITIVITY --------MATRIX THAT ARE ASSOCIATED WITH DRESP2 ENTRIES -------------------------------------------------------------COLUMN DRESP2 SUB FREQ/ NO. ENTRY ID CASE TIME ---------------------------------------------------------86 5 1 87 6 1 88 2401 1 89 5 2 90 6 2 91 2401 2 92 4 6 93 4 6 94 4 6 95 4 6 96 4 6 97 4 6 98 4 6 99 4 6 100 4 6 101 4 6 102 4 6 103 4 6 104 21 7 105 21 7 106 21 7 107 21 7 108 21 7 109 21 7 110 21 7 111 21 7 112 21 7 113 21 7 114 21 7 115 21 7 ^^^ DMAP INFORMATION MESSAGE 9029 (DESOPT) DESIGN SENSITIVITY COEFFICIENT MATRIX FOR DIRECT AND SYNTHETIC RESPONSES - GRADIENTS OF RESPONSES WITH RESPECT TO INDEPENDENT DESIGN VARIABLES MATRIX DSCM2 (GINO NAME 101 ) IS A REAL 115 COLUMN X 3 ROW RECTANG MATRIX. COLUMN 1 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 6.6667E+02 1.3333E+03 5.0000E+01 COLUMN 2 ROWS 1 THRU 3 --------------------------------------------------

8-22

NX Nastran Aeroelastic Analysis User’s Guide

Aeroelastic Design Sensitivities and Optimization

ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 3) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 3) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW

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NX Nastran Aeroelastic Analysis User’s Guide

8-23

Chapter 8

1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1) COLUMN ROW 1)

8-24

Aeroelastic Design Sensitivities and Optimization

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ROW 1) 1.0853E-01 1.8817E-01 1.0261E-02 COLUMN 94 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 1.1232E-01 1.9120E-01 9.6156E-03 COLUMN 95 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -5.1115E-01 -3.9271E-01 -1.1173E-01 COLUMN 96 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -1.8500E+01 -8.8924E+00 -1.6722E+00 COLUMN 97 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 2.1129E+01 1.3226E+01 2.3118E+00 COLUMN 98 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 4.0044E-03 4.5097E-02 9.6770E-04 COLUMN 99 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 3.4179E+01 1.5957E+01 2.7039E+00 COLUMN 100 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -5.9864E+00 -2.4313E+00 -5.0989E-01 COLUMN 101 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 9.2822E-01 7.2692E-01 5.5682E-02 COLUMN 102 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 5.8837E-03 5.9353E-02 1.3099E-03 COLUMN 103 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 7.5407E-03 6.9683E-02 1.5587E-03 COLUMN 104 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 7.5805E-02 1.3940E-01 9.4811E-03 COLUMN 105 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 8.0688E-02 1.4363E-01 8.7182E-03 COLUMN 106 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 8.4733E-02 1.4732E-01 8.0914E-03 COLUMN 107 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -6.5435E-02 9.0995E-02 -1.7362E-03 COLUMN 108 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -1.3020E+00 -6.6887E-01 -5.7264E-02 COLUMN 109 ROWS 1 THRU 3 -------------------------------------------------ROW 1) -7.6806E+00 -3.1820E+00 -2.5453E-01 COLUMN 110 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 5.2690E-03 3.3642E-02 8.6986E-04 COLUMN 111 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 1.4705E+00 1.4853E+00 1.0002E-01 COLUMN 112 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 1.1036E+01 5.7482E+00 3.5277E-01 COLUMN 113 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 5.7897E-01 7.2965E-01 2.5005E-02 COLUMN 114 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 9.0316E-03 6.3687E-02 1.4559E-03 COLUMN 115 ROWS 1 THRU 3 -------------------------------------------------ROW 1) 1.2167E-02 8.5035E-02 1.8582E-03 THE NUMBER OF NON-ZERO TERMS IN THE DENSEST COLUMN = 3 THE DENSITY OF THIS MATRIX IS 98.84 PERCENT. *************************************************************** S U M M A R Y O F D E S I G N C Y C L E H I S T O R Y *************************************************************** NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 1 NUMBER OF OPTIMIZATIONS W.R.T. APPROXIMATE MODELS 0 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY -----------------------------------------------------------------------------------------------------OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT -------------------------------------------------------------------------------------------------------INITIAL 2.005000E+03 1.460522E+01 -------------------------------------------------------------------------------------------------------DESIGN VARIABLE HISTORY -------------------------------------------------------------------------------------------------------INTERNAL | EXTERNAL | | DV. ID. | DV. ID. | LABEL | INITIAL : 1 : 2 : 3 : 4 : 5 : -------------------------------------------------------------------------------------------------------1 | 10 | PBAR101 | 1.0000E+00 : 2 | 20 | PBAR102 | 1.0000E+00 : 3 | 30 | PBAR103 | 1.0000E-01 : -------------------------------------------------------------------------------------------------------*** USER INFORMATION MESSAGE 6464 (DOM12E)

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RUN TERMINATED DUE TO PARAMETER OPTEXIT =

4.

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Listing 8-3. Output for Design Sensitivities of FSW Airplane

8.3

Aeroelastic Optimization of FSW Airplane (Example HA200B)

This example illustrates the aeroelastic optimization features of SOLution 200 when aeroelastic constraints are included along with stress and deflection constraints. The preceding Example HA200A illustrates the design sensitivity calculations. The Bulk Data input files are the same in the two examples except that the PARAM,OPTEXIT,4, PARAM,CDIF = YES, and DSCREEN entries have been deleted, the DELB parameter has been set to its default value, and the subtitle has been changed to reflect the optimization task. The design flight conditions and constraints have been discussed thoroughly in Example HA200A . It is the sequence of iterations in the optimization process leading to the minimum weight optimized design that is of interest here. Figure 8-3 and Figure 8-4 present iteration history data for the design task. It is seen that the weight objective first increases to satisfy the design requirements and then converges rapidly to the final design. Figure 8-4, shows that the design requirements and objectives are met by increasing the thickness of the inboard bar significantly; the outboard bar decreases slightly, and the fin structure, which is relatively unimportant, increases.

Figure 8-3. Convergence of Design Objective (Structural Weight)

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Figure 8-4. Convergence of Design Variables Listing 10-4 shows selected output from the optimization results file. A description of design optimization output can be found in Chapter 5 of the NX Nastran User’s Guide for Design Sensitivity and Optimization . The abridged listing starts with a statement that the maximum constraint is 14.605. It can be determined that this constraint is associated with a flutter damping response that is equal to 0.4382, which is a very large value. The output that follows in Listing 10-2 is produced with P2 = 15 on the DOPTPRM entry and shows the results from the first optimization design cycle. In the series of tables included in this output, INPUT VALUE refers to exact values determined from the analysis. OUTPUT VALUES refers to approximate values as determined by the optimizer. In many cases, there is no approximation or the approximation is exact so that the output values are very accurate. Other values are determined from linear approximations, and an exact analysis could produce a significantly different value. The flutter damping response is a good example where a linear prediction can be expected to become less accurate as the design moves away from the starting point. The results indicate that the weight increased from 2000.5 lbs to 3527.9 lbs for the first design cycle and that the design variables also increased significantly. The DESIGNED PROPERTIES output shows how the design variable changes affect the underlying structural properties. The DESIGN CONSTRAINTS ON RESPONSES is useful in that it gives some insight into which responses are driving the design. Because the DESIGN CONSTRAINT ON RESPONSES table lists responses by internal ID, it is necessary to refer to a second table to identify which responses are impacting the design. In this case, all the constrained responses are synthetic since EQUA is the RESPONSE TYPE for all of the constraints. This indicates that the constraints imposed by the stress limits have been screened out for this design cycle. The table on CONSTRAINTS ON DESIGNED PROPERTIES follows and indicates that in the process of reducing the constraint violation on the responses, certain property limits were exceeded. No problem occurred in this case, but too severe a property constraint violation could result in a meaningless structure (e.g., one with negative bar areas). If this does occur, the user could restart the design from a

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better starting point or tighten the property move limits so that the properties are less likely to become meaningless. The output next contains a series of tables that list the response values ordered by subcase. The response with an internal ID of 10 is the flutter damping value with the large violation of the design requirement. The DRESP2 responses indicate that the roll performance of the initial design is slightly deficient and predict that the designed structure will be adequate in this regard.

Convergence Check Following the design information is a convergence check summary that always appears following an optimization cycle. The table is SOFT CONVERGENCE DECISION LOGIC, and soft convergence refers to a check of the results following an approximate optimization but before a re-analysis has been performed. In this case, it is seen that the convergence criteria all fail in that the objective, property values, and constraints all indicate that further progress is required.

Final Design Results The output shown in Listing 10-4 now skips ahead to the results from the seventh and final design cycle. The results tables show identical input and output values, indicating that the optimizer made no progress for this design cycle and implying that an optimal design has been achieved. The DESIGNED CONSTRAINTS table shows that the maximum constraint value is -0.00263, which indicates that the design is feasible (i.e., the design satisfies the design constraint conditions). The RETAINED DRESP2 RESPONSES table indicates that the roll response requirements are being satisfied almost exactly and that one of the flutter constraints is also at the prescribed limit. The soft convergence check indicates that the design objective and properties have not changed and the maximum constraint is below the prescribed threshold so that all the convergence criteria are satisfied. A final analysis then occurs and the subsonic flutter summary is shown in the listing. Attention is directed to POINT 6 in the summary where it is seen that the flutter curve crosses the zero damping boundary right at 1500.0 ft/sec. Figure 8-5 compares the V-g curves of the initial and final designs, indicating the subsonic flutter velocity has been increased from 1360.0 to 1500.0 ft/sec.

Design Histories Listing 8-4 is concluded with two tables that present the design cycle histories depicted in Figure 8-3 and Figure 8-4. The first table shows the objective function and the maximum constraint value for each of the design cycles. It is seen that the objective increases from 2005.0 to 2601.8 pounds while the maximum constraint decreases from a value of 14.605 to an essentially zero value. The second table presents a history of the design variables and shows that the weight increase can be attributed to more than doubling the structural properties of the inboard bar while making less significant changes in the remaining design variables.

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Figure 8-5. The Critical V-g-V-f Curves for the Forward Swept Wing Aeroelastic Optimization Example

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EXAMPLE HA200B: 30 DEG FWD SWEPT WING WITH CANARD AND FIN PAGE 159 DEMONSTRATION OF AEROELASTIC OPTIMIZATION SUPERSONIC FLUTTER ANALYSIS SUBCASE 7 THIS IS THE FIRST ANALYSIS - NO CONVERGENCE CHECK ------------------------------------------------------MAXIMUM VALUE OF CONSTRAINTS : 1.4605E+01 ------------------------------------------------------******************************************************************** * * * * * D E S I G N O P T I M I Z A T I O N * * * * * ******************************************************************** ******************************************* * * * D E S I G N C Y C L E 1 * * * ******************************************* ***** OPTIMIZATION RESULTS BASED ON THE APPROXIMATE MODEL ***** ----DESIGN OBJECTIVE ------------------------------------------------------------------------------------------------------INTERNAL TYPE MINIMIZE RESPONSE OF OR SUPERELEMENT SUBCASE INPUT OUTPUT ID RESPONSE LABEL MAXIMIZE ID ID VALUE VALUE --------------------------------------------------------------------------------------------------1 DRESP1 WEIGHT MINIMIZE 0 1 2.0050E+03 3.5277E+03 ----DESIGN VARIABLES ------------------------------------------------------------------------------------------------------------INTERNAL DESVAR LOWER INPUT OUTPUT UPPER ID ID LABEL BOUND VALUE VALUE BOUND ---------------------------------------------------------------------------------------------------------1 10 PBAR101 1.0000E-03 1.0000E+00 2.0000E+00 1.0000E+02 2 20 PBAR102 1.0000E-03 1.0000E+00 1.6390E+00 1.0000E+02 3 30 PBAR103 1.0000E-03 1.0000E-01 2.0000E-01 1.0000E+02 ----DESIGNED PROPERTIES --------------------------------------------------------------------------------------------------------------PROPERTY PROPERTY FIELD TYPE OF LOWER INPUT OUTPUT UPPER TYPE ID ID PROPERTY BOUND VALUE VALUE BOUND ---------------------------------------------------------------------------------------------------------PBAR 101 4 DVPREL1 1.0000E-03 1.5000E+00 3.0000E+00 1.0000E+20 PBAR 101 5 DVPREL1 1.0000E-03 1.7361E-01 3.4722E-01 1.0000E+20 PBAR 101 6 DVPREL1 1.0000E-03 2.0000E+00 4.0000E+00 1.0000E+20 PBAR 101 7 DVPREL1 1.0000E-03 4.6296E-01 9.2593E-01 1.0000E+20 PBAR 102 4 DVPREL1 1.0000E-03 1.5000E+00 2.4586E+00 1.0000E+20 PBAR 102 5 DVPREL1 1.0000E-03 1.7361E-01 2.8455E-01 1.0000E+20 PBAR 102 6 DVPREL1 1.0000E-03 2.0000E+00 3.2781E+00 1.0000E+20 PBAR 102 7 DVPREL1 1.0000E-03 4.6296E-01 7.5881E-01 1.0000E+20 PBAR 103 4 DVPREL1 1.0000E-03 1.5000E-01 3.0000E-01 1.0000E+20 PBAR 103 5 DVPREL1 1.0000E-03 1.7361E-02 3.4722E-02 1.0000E+20 PBAR 103 6 DVPREL1 1.0000E-03 2.0000E-01 4.0000E-01 1.0000E+20 PBAR 103 7 DVPREL1 1.0000E-03 4.6296E-02 9.2593E-02 1.0000E+20 ----DESIGN CONSTRAINTS ON RESPONSES ----(MAXIMUM RESPONSE CONSTRAINTS MARKED WITH **) --------------------------------------------------------------------------------------------------------INTERNAL INTERNAL INTERNAL DCONSTR RESPONSE RESPONSE L/U REGION SUBCASE INPUT OUTPUT ID ID ID TYPE FLAG ID ID VALUE VALUE ---------------------------------------------------------------------------------------------------------1 50 2 EQUA UPPER 6 1 -3.9543E-01 -6.6547E-01 2 50 1 EQUA UPPER 5 1 -3.9543E-01 -6.6440E-01 3 50 3 EQUA LOWER 2401 1 1.7848E-02 -3.4503E-02 4 60 4 EQUA LOWER 2401 2 1.1241E-02 -3.9465E-02 5 6 5 EQUA UPPER 4 6 -4.3507E-01 -3.6965E-01 6 6 6 EQUA UPPER 4 6 1.4605E+01** 1.3839E+00** 7 7 7 EQUA UPPER 21 7 8.1231E+00 -8.8540E+00 8 7 8 EQUA UPPER 21 7 -4.7149E-01 -4.1885E-01 ----CONSTRAINTS ON DESIGNED PROPERTIES ------------------------------------------------------------------------------------------------------------INTERNAL PROPERTY FIELD L/U CYCLE INPUT OUTPUT ID ID FLAG FLAG LIMIT VALUE VALUE --------------------------------------------------------------------------------------------------------9 101 4 LOWER 1.2000E+00 -2.5000E-01 -1.5000E+00 10 101 5 LOWER 1.3889E-01 -2.5000E-01 -1.5000E+00 11 101 6 LOWER 1.6000E+00 -2.5000E-01 -1.5000E+00 12 101 7 LOWER 3.7037E-01 -2.5000E-01 -1.5000E+00 13 102 4 LOWER 1.2000E+00 -2.5000E-01 -1.0488E+00 ----CONSTRAINTS ON DESIGNED PROPERTIES -------------------------------------------------------------------------------------------------------------INTERNAL PROPERTY FIELD L/U CYCLE INPUT OUTPUT ID ID FLAG FLAG LIMIT VALUE VALUE --------------------------------------------------------------------------------------------------------14 102 5 LOWER 1.3889E-01 -2.5000E-01 -1.0488E+00 15 102 6 LOWER 1.6000E+00 -2.5000E-01 -1.0488E+00 16 102 7 LOWER 3.7037E-01 -2.5000E-01 -1.0488E+00 17 103 4 LOWER 1.2000E-01 -2.5000E-01 -1.5000E+00 18 103 5 LOWER 7.3611E-03 -1.3585E+00 -3.7170E+00 19 103 6 LOWER 1.6000E-01 -2.5000E-01 -1.5000E+00 20 103 7 LOWER 3.6296E-02 -2.7551E-01 -1.5510E+00 21 101 4 UPPER 1.8000E+00 -1.6667E-01 6.6667E-01 22 101 5 UPPER 2.0833E-01 -1.6667E-01 6.6667E-01

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23 24 25 26 27 28 29 30 31 32

101 101 102 102 102 102 103 103 103 103

6 UPPER 2.4000E+00 -1.6667E-01 6.6667E-01 7 UPPER 5.5556E-01 -1.6667E-01 6.6667E-01 4 UPPER 1.8000E+00 -1.6667E-01 3.6586E-01 5 UPPER 2.0833E-01 -1.6667E-01 3.6586E-01 6 UPPER 2.4000E+00 -1.6667E-01 3.6586E-01 7 UPPER 5.5556E-01 -1.6667E-01 3.6586E-01 4 UPPER 1.8000E-01 -1.6667E-01 6.6667E-01 5 UPPER 2.7361E-02 -3.6548E-01 2.6904E-01 6 UPPER 2.4000E-01 -1.6667E-01 6.6667E-01 7 UPPER 5.6296E-02 -1.7763E-01 6.4474E-01 ---------------------------------------------------------------------| R E S P O N S E S IN D E S I G N M O D E L | ---------------------------------------------------------------------(N/A - BOUND NOT ACTIVE OR AVAILABLE) ----WEIGHT RESPONSE --------------------------------------------------------------------------------------------------------INTERNAL DRESP1 RESPONSE LOWER INPUT OUTPUT UPPER ID ID LABEL BOUND VALUE VALUE BOUND ----------------------------------------------------------------------------------------------------1 10 WEIGHT N/A 2.0050E+03 3.5277E+03 N/A D E S I G N C Y C L E = 1 S U B C A S E = 1 ----DISPLACEMENT RESPONSES ----------------------------------------------------------------------------------------------------INTERNAL DRESP1 RESPONSE GRID COMPONENT LOWER INPUT OUTPUT UPPER ID ID LABEL ID NO. BOUND VALUE VALUE BOUND ---------------------------------------------------------------------------------------------------2 100 RTROT 100 5 N/A 1.1291E-02 1.1403E-02 N/A 3 101 RTIPROT 120 5 N/A 2.1841E-02 1.7259E-02 N/A 4 201 LTIPROT 220 5 N/A 2.1841E-02 1.7240E-02 N/A ----STABILITY DERIVATIVE RESPONSES -------------------------------------------------------------------------------------------------------RESTRAINED / AESTAT / INTERNAL DRESP1 RESPONSE UNRESTRAINED COMPONENT AESURF LOWER INPUT OUTPUT UPPER ID ID LABEL FLAG NO. ID BOUND VALUE VALUE BOUND ------------------------------------------------------------------------------------------------------5 1401 CLDELTA 0 4 517 N/A 2.6340E-01 2.6922E-01 N/A 6 1402 CLP 0 4 513 N/A -4.4697E-01 -4.3374E-01 N/A D E S I G N C Y C L E = 1 S U B C A S E = 2 ----STABILITY DERIVATIVE RESPONSES -------------------------------------------------------------------------------------------------------------RESTRAINED / AESTAT / INTERNAL DRESP1 RESPONSE UNRESTRAINED COMPONENT AESURF LOWER INPUT OUTPUT UPPER ID ID LABEL FLAG NO. ID BOUND VALUE VALUE BOUND ----------------------------------------------------------------------------------------------------------7 1401 CLDELTA 0 4 517 N/A 2.1655E-01 2.2531E-01 N/A 8 1402 CLP 0 4 513 N/A -5.0933E-01 -5.0408E-01 N/A D E S I G N C Y C L E = 1 S U B C A S E = 6 ----FLUTTER RESPONSES - -------------------------------------------------------------------------------------------------------------INTERNAL DRESP1 RESPONSE MODE MACH LOWER INPUT OUTPUT UPPER ID ID LABEL NO. VELOCITY NO. BOUND VALUE VALUE BOUND ---------------------------------------------------------------------------------------------------------9 1 FLUTTER 6 1.0000E+03 9.0000E-01 N/A -1.3052E-02 -1.1089E-02 N/A 10 1 FLUTTER 6 1.5000E+03 9.0000E-01 N/A 4.3816E-01 4.1516E-02 N/A D E S I G N C Y C L E = 1 S U B C A S E = 7 ----FLUTTER RESPONSES -----------------------------------------------------------------------------------------------------------INTERNAL DRESP1 RESPONSE MODE MACH LOWER INPUT OUTPUT UPPER ID ID LABEL NO. VELOCITY NO. BOUND VALUE VALUE BOUND -------------------------------------------------------------------------------------------------------11 11 FLUTTER 5 1.5000E+03 1.2000E+00 N/A 2.4369E-01 -2.6562E-01 N/A 12 11 FLUTTER 6 1.0000E+03 1.2000E+00 N/A -1.4145E-02 -1.2566E-02 N/A ---- RETAINED DRESP2 RESPONSES ----------------------------------------------------------------------------------------------------------INTERNAL DRESP2 RESPONSE EQUATION LOWER INPUT OUTPUT UPPER ID ID LABEL ID BOUND ALUE VALUE BOUND --------------------------------------------------------------------------------------------------------1 5 RHSTWIST 5 N/A 1.0550E-02 5.8562E-03 1.7450E-02 2 6 LHSTWIST 5 N/A 1.0550E-02 5.8376E-03 1.7450E-02 3 2401 ROLLEFF 103 6.0000E-01 5.8929E-01 6.2070E-01 N/A 4 2401 ROLLEFF 103 4.3000E-01 4.2517E-01 4.4697E-01 N/A 5 4 GDAMP 4 N/A -4.3052E-01 -4.1089E-01 -3.0000E-01 6 4 GDAMP 4 N/A 4.0816E+00 1.1516E-01 -3.0000E-01 7 21 GDAMP 4 N/A 2.1369E+00 -2.9562E+00 -3.0000E-01 8 21 GDAMP 4 N/A -4.4145E-01 -4.2566E-01 -3.0000E-01 **************************************************************************************** INSPECTION OF CONVERGENCE DATA FOR THE OPTIMAL DESIGN WITH RESPECT TO APPROXIMATE MODELS (SOFT CONVERGENCE DECISION LOGIC) **************************************************************************************** RELATIVE CHANGE IN OBJECTIVE 7.5947E-01 MUST BE LESS THAN 1.0000E-03 OR ABSOLUTE CHANGE IN OBJECTIVE 1.5227E+03 MUST BE LESS THAN 1.0000E-02 AND MAX OF RELATIVE PROP.CHANGES 1.0000E+00 MUST BE LESS THAN 1.0000E-03 --- AND --MAXIMUM CONSTRAINT VALUE 1.3839E+00 MUST BE LESS THAN 5.0000E-03 OR MAX OF RELATIVE D.V. CHANGES 1.0000E+00 MUST BE LESS THAN 1.0000E-03 **************************************************************************************** . . . Note: FOUR HUNDRED pages of output have been deleted.

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Chapter 8

Aeroelastic Design Sensitivities and Optimization

. . . ******************************************************************** * * * * * D E S I G N O P T I M I Z A T I O N * * * * * ******************************************************************** ******************************************* * * * D E S I G N C Y C L E 7 * * * ******************************************* ***** OPTIMIZATION RESULTS BASED ON THE APPROXIMATE MODEL ***** ----DESIGN OBJECTIVE ------------------------------------------------------------------------------------------------------------INTERNAL TYPE MINIMIZE RESPONSE OF OR SUPERELEMENT SUBCASE INPUT OUTPUT ID RESPONSE LABEL MAXIMIZE ID ID VALUE VALUE --------------------------------------------------------------------------------------------------------1 DRESP1 WEIGHT MINIMIZE 0 1 2.6018E+03 2.6018E+03 ----DESIGN VARIABLES ------------------------------------------------------------------------------------------------------------INTERNAL DESVAR LOWER INPUT OUTPUT UPPER ID ID LABEL BOUND VALUE VALUE BOUND --------------------------------------------------------------------------------------------------------1 10 PBAR101 1.0000E-03 2.0927E+00 2.0927E+00 1.0000E+02 2 20 PBAR102 1.0000E-03 8.9963E-01 8.9963E-01 1.0000E+02 3 30 PBAR103 1.0000E-03 1.4322E-01 1.4322E-01 1.0000E+02 ----DESIGNED PROPERTIES -----------------------------------------------------------------------------------------------------------PROPERTY PROPERTY FIELD TYPE OF LOWER INPUT OUTPUT UPPER TYPE ID ID PROPERTY BOUND VALUE VALUE BOUND --------------------------------------------------------------------------------------------------------PBAR 101 4 DVPREL1 1.0000E-03 3.1391E+00 3.1391E+00 1.0000E+20 PBAR 101 5 DVPREL1 1.0000E-03 3.6332E-01 3.6332E-01 1.0000E+20 PBAR 101 6 DVPREL1 1.0000E-03 4.1855E+00 4.1855E+00 1.0000E+20 PBAR 101 7 DVPREL1 1.0000E-03 9.6886E-01 9.6886E-01 1.0000E+20 PBAR 102 4 DVPREL1 1.0000E-03 1.3494E+00 1.3494E+00 1.0000E+20 PBAR 102 5 DVPREL1 1.0000E-03 1.5619E-01 1.5619E-01 1.0000E+20 PBAR 102 6 DVPREL1 1.0000E-03 1.7993E+00 1.7993E+00 1.0000E+20 PBAR 102 7 DVPREL1 1.0000E-03 4.1649E-01 4.1649E-01 1.0000E+20 PBAR 103 4 DVPREL1 1.0000E-03 2.1482E-01 2.1482E-01 1.0000E+20 PBAR 103 5 DVPREL1 1.0000E-03 2.4864E-02 2.4864E-02 1.0000E+20 PBAR 103 6 DVPREL1 1.0000E-03 2.8643E-01 2.8643E-01 1.0000E+20 PBAR 103 7 DVPREL1 1.0000E-03 6.6304E-02 6.6304E-02 1.0000E+20 ----DESIGN CONSTRAINTS ON RESPONSES ----(MAXIMUM RESPONSE CONSTRAINTS MARKED WITH **) -------------------------------------------------------------------------------------------------------INTERNAL INTERNAL INTERNAL DCONSTR RESPONSE RESPONSE L/U REGION SUBCASE INPUT OUTPUT ID ID ID TYPE FLAG ID ID VALUE VALUE -------------------------------------------------------------------------------------------------------1 50 1 EQUA LOWER 2401 1 -6.4307E-03 -6.4307E-03 2 60 2 EQUA LOWER 2401 2 -1.5090E-02 -1.5090E-02 3 6 3 EQUA UPPER 4 6 -4.4682E-01 -4.4682E-01 4 6 4 EQUA UPPER 4 6 -2.5652E-03** -2.5652E-03** 5 7 5 EQUA UPPER 21 7 -4.7816E-01 -4.7816E-01 --------CONSTRAINTS ON DESIGNED PROPERTIES --------------------------------------------------------------------------------------------------------CYCLE INPUT OUTPUT INTERNAL PROPERTY FIELD L/U ID ID FLAG FLAG LIMIT VALUE VALUE --------------------------------------------------------------------------------------------------------6 101 4 LOWER 2.9821E+00 -5.2632E-02 -5.2632E-02 -5.2632E-02 7 101 5 LOWER 3.4516E-01 -5.2632E-02 8 101 6 LOWER 3.9762E+00 -5.2632E-02 -5.2632E-02 -5.2632E-02 9 101 7 LOWER 9.2041E-01 -5.2632E-02 10 102 4 LOWER 1.2820E+00 -5.2632E-02 -5.2632E-02 -5.2632E-02 11 102 5 LOWER 1.4838E-01 -5.2632E-02 12 102 6 LOWER 1.7093E+00 -5.2632E-02 -5.2632E-02 -5.2632E-02 13 102 7 LOWER 3.9567E-01 -5.2632E-02 14 103 4 LOWER 2.0408E-01 -5.2632E-02 -5.2632E-02 -1.1179E-01 15 103 5 LOWER 2.2364E-02 -1.1179E-01 16 103 6 LOWER 2.7211E-01 -5.2632E-02 -5.2632E-02 -5.2632E-02 17 103 7 LOWER 6.2989E-02 -5.2632E-02 18 101 4 UPPER 3.2961E+00 -4.7619E-02 -4.7619E-02 -4.7619E-02 19 101 5 UPPER 3.8149E-01 -4.7619E-02 20 101 6 UPPER 4.3947E+00 -4.7619E-02 -4.7619E-02 -4.7619E-02 21 101 7 UPPER 1.0173E+00 -4.7619E-02 22 102 4 UPPER 1.4169E+00 -4.7619E-02 -4.7619E-02 -4.7619E-02 23 102 5 UPPER 1.6399E-01 -4.7619E-02 24 102 6 UPPER 1.8892E+00 -4.7619E-02 -4.7619E-02 -4.7619E-02 25 102 7 UPPER 4.3732E-01 -4.7619E-02 26 103 4 UPPER 2.2557E-01 -4.7619E-02 -4.7619E-02 -9.1361E-02 27 103 5 UPPER 2.7364E-02 -9.1361E-02 28 103 6 UPPER 3.0075E-01 -4.7619E-02 -4.7619E-02 -4.7619E-02 29 103 7 UPPER 6.9619E-02 -4.7619E-02 ----------------------------------------------------------------------

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NX Nastran Aeroelastic Analysis User’s Guide

Aeroelastic Design Sensitivities and Optimization

| R E S P O N S E S IN D E S I G N M O D E L | ---------------------------------------------------------------------(N/A - BOUND NOT ACTIVE OR AVAILABLE) --------WEIGHT RESPONSE -------------------------------------------------------------------------------------------------------LOWER INPUT OUTPUT UPPER INTERNAL DRESP1 RESPONSE ID ID LABEL BOUND VALUE VALUE BOUND --------------------------------------------------------------------------------------------------------1 10 WEIGHT N/A 2.6018E+03 2.6018E+03 N/A 1 D E S I G N C Y C L E = 7 S U B C A S E = ----STABILITY DERIVATIVE RESPONSES ------------------------------------------------------------------------------------------------------------RESTRAINED / AESTAT / LOWER INPUT OUTPUT UPPER INTERNAL DRESP1 RESPONSE UNRESTRAINED COMPONENT AESURF ID ID LABEL FLAG NO. ID BOUND VALUE VALUE BOUND ---------------------------------------------------------------------------------------------------------2 1401 CLDELTA 0 4 517 N/A 2.6580E-01 2.6580E-01 N/A -4.4017E-01 N/A 3 1402 CLP 0 4 513 N/A -4.4017E-01 D E S I G N C Y C L E = 7 S U B C A S E = 2 --------STABILITY DERIVATIVE RESPONSES --------------------------------------------------------------------------------------------------------RESTRAINED / AESTAT / INTERNAL DRESP1 RESPONSE UNRESTRAINED COMPONENT AESURF LOWER INPUT OUTPUT UPPER VALUE VALUE BOUND ID ID LABEL FLAG NO. ID BOUND --------------------------------------------------------------------------------------------------------2.2095E-01 2.2095E-01 N/A 4 1401 CLDELTA 0 4 517 N/A 5 1402 CLP 0 4 513 N/A -5.0621E-01 -5.0621E-01 N/A 6 D E S I G N C Y C L E = 7 S U B C A S E = ----FLUTTER RESPONSES -------------------------------------------------------------------------------------------------------------INTERNAL DRESP1 RESPONSE MODE MACH LOWER INPUT OUTPUT UPPER NO. BOUND VALUE VALUE BOUND ID ID LABEL NO. VELOCITY --------------------------------------------------------------------------------------------------------9.0000E-01 N/A -1.3405E-02 -1.3405E-02 N/A 6 1 FLUTTER 6 1.0000E+03 7 1 FLUTTER 6 1.5000E+03 9.0000E-01 N/A -7.6959E-05 -7.6959E-05 N/A 7 D E S I G N C Y C L E = 7 S U B C A S E = ----FLUTTER RESPONSES ------------------------------------------------------------------------------------------------------------INTERNAL DRESP1 RESPONSE MODE MACH LOWER INPUT OUTPUT UPPER BOUND VALUE VALUE BOUND ID ID LABEL NO. VELOCITY NO. --------------------------------------------------------------------------------------------------------1.2000E+00 N/A -1.4345E-02 -1.4345E-02 N/A 8 11 FLUTTER 6 1.0000E+03 ---- RETAINED DRESP2 RESPONSES ----------------------------------------------------------------------------------------------------------INTERNAL DRESP2 RESPONSE EQUATION LOWER INPUT OUTPUT UPPER ID ID LABEL ID BOUND VALUE VALUE BOUND -------------------------------------------------------------------------------------------------------1 2401 ROLLEFF 103 6.0000E-01 6.0386E-01 6.0386E-01 N/A 2 2401 ROLLEFF 103 4.3000E-01 4.3649E-01 4.3649E-01 N/A 3 4 GDAMP 4 N/A -4.3405E-01 -4.3405E-01 -3.0000E-01 4 4 GDAMP 4 N/A -3.0077E-01 -3.0077E-01 -3.0000E-01 5 21 GDAMP 4 N/A -4.4345E-01 -4.4345E-01 -3.0000E-01 ************************************************** TERMINATION OF DESIGN ITERATION (SOFT CONVERGENCE) ************************************************** RELATIVE CHANGE IN OBJECTIVE 0.0000E+00 MUST BE LESS THAN 1.0000E-03 OR ABSOLUTE CHANGE IN OBJECTIVE 0.0000E+00 MUST BE LESS THAN 1.0000E-02 AND MAXIMUM OF RELATIVE PROP. CHANGES 0.0000E+00 MUST BE LESS THAN 1.0000E-03 --- AND --MAXIMUM CONSTRAINT VALUE -2.5652E-03 MUST BE LESS THAN 5.0000E-03 OR MAXIMUM OF RELATIVE D.V. CHANGES 0.0000E+00 MUST BE LESS THAN 1.0000E-03 EXPLANATION: THE OPTIMIZATION PROCESS WITH RESPECT TO THE APPROXIMATE MODELS DID NOT CHANGE APPRECIABLY. THIS DESIGN MAY NOT WARRANT AN ADDITIONAL COMPLETE FINITE ELEMENT ANALYSIS. ^^^ DMAP INFORMATION MESSAGE 9024 (FEA,PHASE1) STIFFNESS, MASS, DAMPING, AND LOAD GENERATION INITIATED. DESIGN CYCLE NUMBER= 8 EXAMPLE HA200B: 30 DEG FWD SWEPT WING WITH CANARD AND FIN DEMONSTRATION OF AEROELASTIC OPTIMIZATION SUBSONIC FLUTTER ANALYSIS FLUTTER SUMMARY POINT = 1 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.0000 9.9999996E+24 1.0000000E+03 -1.1597837E-07 0.0000000E+00 0.0000 9.9999996E+24 1.1000000E+03 -1.1662188E-07 0.0000000E+00 0.0000 9.9999996E+24 1.2000000E+03 -1.1752252E-07 0.0000000E+00 0.0000 9.9999996E+24 1.3000000E+03 -1.1829093E-07 0.0000000E+00 0.0000 9.9999996E+24 1.4000000E+03 -1.1912391E-07 0.0000000E+00 0.0000 9.9999996E+24 1.5000000E+03 -1.2002239E-07 0.0000000E+00 0.0000 9.9999996E+24 1.6000000E+03 -1.2098708E-07 0.0000000E+00 POINT = 2 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY 0.0000 9.9999996E+24 1.0000000E+03 1.1691864E-05 0.0000000E+00 0.0000 9.9999996E+24 1.1000000E+03 1.1695184E-05 0.0000000E+00 0.0000 9.9999996E+24 1.2000000E+03 1.1711473E-05 0.0000000E+00 0.0000 9.9999996E+24 1.3000000E+03 1.1714760E-05 0.0000000E+00 0.0000 9.9999996E+24 1.4000000E+03 1.1718438E-05 0.0000000E+00 0.0000 9.9999996E+24 1.5000000E+03 1.1722215E-05 0.0000000E+00 0.0000 9.9999996E+24 1.6000000E+03 1.1726737E-05 0.0000000E+00 POINT = 3 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY

PAGE

693

SUBCASE 6 METHOD = PK COMPLEX -8.0389236E-06 -8.8918814E-06 -9.7751463E-06 -1.0658982E-05 -1.1559736E-05 -1.2478847E-05 -1.3417758E-05 METHOD = PK COMPLEX 8.1040978E-04 8.9170400E-04 9.7412284E-04 1.0555959E-03 1.1371524E-03 1.2187704E-03 1.3005233E-03 METHOD = PK COMPLEX

EIGENVALUE 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 EIGENVALUE 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 EIGENVALUE

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Chapter 8

Aeroelastic Design Sensitivities and Optimization

0.0001 1.4836668E+04 1.0000000E+03 7.0408906E-04 2.1454270E-03 4.7456010E-06 1.3480116E-02 0.0001 1.5005404E+04 1.1000000E+03 6.3301768E-04 2.3334317E-03 4.6404580E-06 1.4661385E-02 0.0001 1.5193557E+04 1.2000000E+03 6.2505697E-04 2.5140385E-03 4.9367532E-06 1.5796170E-02 0.0001 1.5402303E+04 1.3000000E+03 6.1492290E-04 2.6866295E-03 5.1901316E-06 1.6880592E-02 0.0001 1.5632923E+04 1.4000000E+03 6.0022436E-04 2.8506110E-03 5.3752847E-06 1.7910918E-02 0.0001 1.5886951E+04 1.5000000E+03 5.1911175E-04 3.0053896E-03 4.9013024E-06 1.8883420E-02 0.0001 1.6166213E+04 1.6000000E+03 5.0864479E-04 3.1503716E-03 5.0341514E-06 1.9794369E-02 POINT = 4 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0365 2.7418503E+01 1.0000000E+03 -2.5639358E-01 1.1609309E+00 -9.3511158E-01 7.2943444E+00 0.0365 2.7424862E+01 1.1000000E+03 -2.5603378E-01 1.2767279E+00 -1.0269411E+00 8.0219183E+00 0.0365 2.7431917E+01 1.2000000E+03 -2.5563699E-01 1.3924359E+00 -1.1182754E+00 8.7489328E+00 0.0364 2.7439695E+01 1.3000000E+03 -2.5520208E-01 1.5080445E+00 -1.2090611E+00 9.4753237E+00 0.0364 2.7448221E+01 1.4000000E+03 -2.5472856E-01 1.6235435E+00 -1.2992462E+00 1.0201025E+01 0.0364 2.7457542E+01 1.5000000E+03 -2.5421497E-01 1.7389205E+00 -1.3887713E+00 1.0925960E+01 0.0364 2.7467678E+01 1.6000000E+03 -2.5366071E-01 1.8541640E+00 -1.4775808E+00 1.1650057E+01 POINT = 5 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.0647 1.5451419E+01 1.0000000E+03 -7.0804781E-01 2.0600691E+00 -4.5824132E+00 1.2943796E+01 0.0690 1.4500205E+01 1.1000000E+03 -6.9783837E-01 2.4147305E+00 -5.2938719E+00 1.5172200E+01 0.0749 1.3352010E+01 1.2000000E+03 -6.8083292E-01 2.8607814E+00 -6.1189251E+00 1.7974821E+01 0.0840 1.1904034E+01 1.3000000E+03 -6.4768982E-01 3.4761567E+00 -7.0732059E+00 2.1841337E+01 0.1019 9.8146553E+00 1.4000000E+03 -5.5107415E-01 4.5404940E+00 -7.8607321E+00 2.8528765E+01 0.1102 9.0704813E+00 1.5000000E+03 -1.3603585E+00 5.2639408E+00 -2.2496464E+01 3.3074318E+01 0.0882 1.1338793E+01 1.6000000E+03 -2.1910040E+00 4.4916224E+00 -3.0916929E+01 2.8221699E+01 POINT = 6 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.2590 3.8605058E+00 1.0000000E+03 -1.3404722E-02 8.2452888E+00 -3.4722710E-01 5.1806683E+01 0.2356 4.2437339E+00 1.1000000E+03 -1.4775367E-02 8.2507734E+00 -3.8298592E-01 5.1841141E+01 0.2162 4.6260843E+00 1.2000000E+03 -1.6175762E-02 8.2569141E+00 -4.1959706E-01 5.1879726E+01 0.1997 5.0074677E+00 1.3000000E+03 -1.7611071E-02 8.2637148E+00 -4.5720500E-01 5.1922451E+01 0.1680 5.9511795E+00 1.4000000E+03 -4.7586706E-01 7.4881597E+00 -1.1194653E+01 4.7049496E+01 0.1176 8.5006313E+00 1.5000000E+03 -7.6958990E-05 5.6168160E+00 -1.3579989E-03 3.5291496E+01 0.1114 8.9728155E+00 1.6000000E+03 3.2491559E-01 5.6759868E+00 5.7937770E+00 3.5663277E+01 POINT = 7 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.3348 2.9864814E+00 1.0000000E+03 -1.6468352E-01 1.0658359E+01 -5.5142994E+00 6.6968445E+01 0.2924 3.4197140E+00 1.1000000E+03 -2.0586941E-01 1.0238894E+01 -6.6220851E+00 6.4332870E+01 0.2530 3.9518635E+00 1.2000000E+03 -2.6156029E-01 9.6656132E+00 -7.9423876E+00 6.0730839E+01 0.2140 4.6735101E+00 1.3000000E+03 -3.3906546E-01 8.8542194E+00 -9.4315643E+00 5.5632706E+01 0.1856 5.3877926E+00 1.4000000E+03 -1.9086335E-02 8.2711763E+00 -4.9595207E-01 5.1969334E+01 0.1734 5.7669630E+00 1.5000000E+03 -2.0605817E-02 8.2793112E+00 -5.3596193E-01 5.2020447E+01 0.1627 6.1448770E+00 1.6000000E+03 -2.2173816E-02 8.2881365E+00 -5.7736075E-01 5.2075901E+01 POINT = 8 MACH NUMBER = 0.9000 DENSITY RATIO = 1.0000E+00 METHOD = PK KFREQ 1./KFREQ VELOCITY DAMPING FREQUENCY COMPLEX EIGENVALUE 0.6025 1.6596277E+00 1.0000000E+03 -2.9496552E-02 1.9179596E+01 -1.7772994E+00 1.2050896E+02 0.5481 1.8245530E+00 1.1000000E+03 -3.1476546E-02 1.9190502E+01 -1.8976815E+00 1.2057748E+02 0.5027 1.9890791E+00 1.2000000E+03 -3.3288341E-02 1.9203451E+01 -2.0082664E+00 1.2065884E+02 0.4644 2.1531701E+00 1.3000000E+03 -3.4974035E-02 1.9218307E+01 -2.1115956E+00 1.2075219E+02 0.4316 2.3168111E+00 1.4000000E+03 -3.6630649E-02 1.9234793E+01 -2.2135127E+00 1.2085577E+02 0.4032 2.4800010E+00 1.5000000E+03 -3.8330588E-02 1.9252605E+01 -2.3183813E+00 1.2096769E+02 0.3784 2.6427841E+00 1.6000000E+03 -4.0120974E-02 1.9271183E+01 -2.4290125E+00 1.2108442E+02 ***** NORMAL CONVERGENCE CRITERIA SATISFIED ***** (HARD CONVERGENCE DECISION LOGIC) ************************************************************************************** CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA (HARD CONVERGENCE DECISION LOGIC) RELATIVE CHANGE IN OBJECTIVE 0.0000E+00 MUST BE LESS THAN 1.0000E-03 OR ABSOLUTE CHANGE IN OBJECTIVE 0.0000E+00 MUST BE LESS THAN 1.0000E-02 --- AND --MAXIMUM CONSTRAINT VALUE -2.5652E-03 MUST BE LESS THAN 5.0000E-03 (CONVERGENCE TO A FEASIBLE DESIGN) --- OR --MAXIMUM OF RELATIVE PROP. CHANGES 0.0000E+00 MUST BE LESS THAN 1.0000E-03 AND MAXIMUM OF RELATIVE D.V. CHANGES 0.0000E+00 MUST BE LESS THAN 1.0000E-03 (CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN) ************************************************************************************** *************************************************************** S U M M A R Y O F D E S I G N C Y C L E H I S T O R Y *************************************************************** (HARD CONVERGENCE ACHIEVED) (SOFT CONVERGENCE ACHIEVED) NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 8 NUMBER OF OPTIMIZATIONS W.R.T. APPROXIMATE MODELS 7 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY --------------------------------------------------------------------------------------------------------OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT -------------------------------------------------------------------------------------------------------INITIAL 2.005000E+03 1.460522E+01 1 3.527728E+03 3.528712E+03 -2.787540E-04 -3.269444E-02 2 2.823344E+03 2.822969E+03 1.327524E-04 -1.713157E-02 3 2.627507E+03 2.627666E+03 -6.067127E-05 7.991234E-02 4 2.591437E+03 2.591402E+03 1.318965E-05 1.907182E-01 5 2.595658E+03 2.595581E+03 2.962894E-05 1.129563E-01 6 2.601817E+03 2.601818E+03 -4.691731E-07 -2.565185E-03 7 2.601818E+03 2.601818E+03 0.000000E+00 -2.565185E-03 ------------------------------------------------------------------------------------------------------------

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NX Nastran Aeroelastic Analysis User’s Guide

Aeroelastic Design Sensitivities and Optimization

DESIGN VARIABLE HISTORY ---------------------------------------------------------------------------------------------------------INTERNAL | EXTERNAL | | DV. ID. | DV. ID. | LABEL | INITIAL : 1 : 2 : 3 : 4 : 5 : --------------------------------------------------------------------------------------------------------1 | 10 | PBAR101 | 1.0000E+00 : 2.0000E+00 : 1.6000E+00 : 1.8339E+00 : 1.9884E+00 : 2.0878E+00 : 2 | 20 | PBAR102 | 1.0000E+00 : 1.6390E+00 : 1.3112E+00 : 1.0490E+00 : 9.4408E-01 : 8.9742E-01 : 3 | 30 | PBAR103 | 1.0000E-01 : 2.0000E-01 : 1.6000E-01 : 1.2795E-01 : 1.4074E-01 : 1.4312E-01 : --------------------------------------------------------------------------------------------------------INTERNAL| EXTERNAL | | DV. ID. | DV. ID. | LABEL | 6 : 7 : 8 : 9 : 10 : 11 : --------------------------------------------------------------------------------------------------------1 | 10 | PBAR101 | 2.0927E+00 : 2.0927E+00 : 2 | 20 | PBAR102 | 8.9963E-01 : 8.9963E-01 : 3 | 30 | PBAR103 | 1.4322E-01 : 1.4322E-01 : ---------------------------------------------------------------------------------------------------------*** USER INFORMATION MESSAGE 6464 (DOM12E) RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER = 7.

NX Nastran Aeroelastic Analysis User’s Guide

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Chapter 8

Aeroelastic Design Sensitivities and Optimization

Listing 8-4. Output from Optimization of FSW Airplane

8-38

NX Nastran Aeroelastic Analysis User’s Guide

Chapter

9 Dynamic Aeroelastic Response Analysis

•

Overview

•

Discrete Gust Response of BAH Wing (Example HA146A)

•

Transient Rolling of BAH Wing Due to Aileron (Example HA146B)

•

Random Gust Response of BAH Wing (Example HA146C)

•

Frequency Response of BAH Wing to Oscillating Aileron (Examples HA146D and HA146DR)

•

Subsonic Transient Response Analysis of a Sweptback Wing to an Impulsive Force Applied at the Tip (Example HA146E)

NX Nastran Aeroelastic Analysis User’s Guide

9-1

Chapter 9

9.1

Dynamic Aeroelastic Response Analysis

Overview

The dynamic aeroelastic response analysis provided in NX Nastran is demonstrated in five applications: four applications use the BAH jet transport wing and one application uses the 15-deg sweptback wing. Examples HA146A , HA146B , and HA146C are analyses of the transient response to a discrete gust, the random response to atmospheric turbulence, and the transient response to an aileron input for the BAH wing, respectively. For the discrete and random gust examples, the gust field is assumed to be uniform spanwise and the aircraft is free to plunge; the pitching and rolling degrees of freedom are constrained. For the aileron response example, motion is antisymmetric and the aircraft is free to roll while constrained in pitch and plunge. The aerodynamic loads on the wing-aileron combination are calculated using the Doublet-Lattice method at a Mach number of zero. Examples HA146D and HA146DR are frequency response analyses of the BAH wing. A frequency response analysis is always done as a first step within the aeroelastic response solution. However, as demonstrated here, it may be done without any subsequent analysis (e.g., random analysis) in the run. Example HA146E provides an analysis of the response of the 15-deg sweptback wing to an impulsively applied load at the wing tip in order to investigate the transient response at subcritical, critical, and supercritical speeds. The structural and aerodynamic models are the same as in the subsonic flutter analysis described in Example HA145E . It should be noted that Case Control is an essential part of Dynamic Aeroelastic Analysis in that any output not requested in the Case Control Section or the Bulk Data Section will not be provided. In Static Aeroelastic Response and Aerodynamic Flutter analyses, certain output is provided by default. There is no default output from Dynamic Aeroelastic Response analysis.

9.2

Discrete Gust Response of BAH Wing (Example HA146A)

This example calculates the time history of the response of the BAH wing due to a gust load. The gust is a discrete, sharp-edged, symmetric, vertical gust. The structural model in this example is the same as the model described in Example HA144B . The structural input file is BAH_STRUCT.DAT and is shown in Listing 6-4; the inertial input file is BAH_MASS and is shown in Listing 6-5. The simplified aerodynamic box characteristics of Example HA145B are used here; the aerodynamic input file is BAH_AERO5.DAT and it is shown in Listing 7-4. The additional input data required by SOL 146 are described below. The data set BAH_STRUCT.DAT provides all of the basic data for this example. It is necessary to alter constraints for the symmetrical condition considered here on SPC 14 to constrain the suport (GRID 11) in roll and pitch. The wing is therefore only free in plunge to permit correlation with the gust response analysis of Bisplinghoff, Ashley, and Halfman (1956, pp. 682-685). It is also important to include structural damping in a gust response analysis. It is included here as modal structural damping via the tabular form in TABDMP1 2000 (along with SDAMP in Case Control). A realistic value of g = 0.02 is assumed to be constant in the frequency range from zero to 10.0 Hz. (Note that in the Fourier analyses in NX Nastran tabular entries are not extrapolated beyond the tabulated range.) The responses are calculated at a velocity of 475 mph (8360. in/s). This velocity is listed on the AERO entry along with the aerodynamic coordinate system that prescribes the positive x-coordinate downstream, the reference semichord, a sea-level density, and the symmetry flag selected for symmetrical motion. The aerodynamic model is INCLUDEd with the input file BAH_AERO5.DAT, which divides the wing into six strips (including one to represent the fuselage

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carry-over loads). The GUST 1000 entry defines the external aerodynamic load that comes from a stationary one-dimensional vertical gust. GUST 1000 defines the magnitude of the gust angle of attack with a scale factor WG = 1.0. It is desired to delay the application of the gust for 0.2 sec in order to observe if the response has damped out sufficiently during the previous period in the periodic Fourier solution. To obtain the 0.2 sec delay, the GUST entry defines the reference location of the gust at 1672 in. (0.2 sec × 8360. in/s) ahead of the aerodynamic coordinate system (x0 = −1672.0 in) and lists the airplane velocity (8360. in/s). The GUST entry also specifies the TLOAD1 1001 entry that defines the gust time history via the TABLED1 1003 entry. A square-edged gust of unit magnitude is defined in the table with a 1.0 second duration. Because the Fourier transform method is used in the NX Nastran transient aeroelastic analysis, the accelerations, internal forces, stresses, and relative displacements characteristically will all approach some constant value (i.e., show some relative response) and relative velocities will approach a value of zero at the end of each period. However, a technique can be used that results in each of these response quantities, except displacements, approaching a value of zero (i.e., show an absolute response). The technique requires following the applied loading with an identical loading but in the opposite direction; the second loading could be delayed but that is not recommended). This technique is illustrated here. To represent the sharp-edged gust, the TABLED1 1003 entry defines a square-wave gust profile of sufficient duration to permit the peak response to occur, followed by the negative square-wave gust, and a sufficiently long period is used so that the response damps out adequately before the next period begins. Another technique, not demonstrated here, would cause all of the response quantities, including displacements, to approach values of zero. This would follow the applied loading with the identical loading twice. The second loading would be in the opposite direction as the first but with twice the magnitude, and the third loading would have the same magnitude and direction as the first. The response quantities determined when using either of these techniques are only valid during the application of the initial loading; they are not valid during or after the application of the second loading. For this reason, the initial loading must have sufficient duration so that all critical responses are achieved. The DAREA 1002 entry called by the TLOAD1 1001 entry has no influence on the load; it is a dummy entry required by the presence of the TLOAD1 entry. However, the entry must specify at least one point, component, and area. Only the DAREA entry associated with the GUST entry is a dummy; if other loads are prescribed in the analysis, DAREA entries associated with these other applied loads would be used. The MKAERO1 entry selects the Mach numbers and reduced frequencies used in generating , the aerodynamic matrices. The Mach number is 0.62. The reduced frequencies, are based on the velocity and reference chord from the AERO entry. At the cut-off frequency of 10.0 Hz, selected here to include the first four modes, and the 475 mph velocity, the reduced frequency is 0.493. Values of k = 0.001, 0.02, 0.10, and 0.50 are specified on the MKAERO1 entry to represent the range of reduced frequencies. The PARAM,GUSTAERO entry is set to -1 in order for the harmonic gust aerodynamic coefficients to be calculated in addition to the aerodynamic coefficients for harmonic motion. When there are prescribed applied loads but no gust loading in the transient response analysis, neither the PARAM,GUSTAERO nor the GUST entries are required. The flight condition is additionally specified by the Mach number on the PARAM,MACH,0.62 entry and dynamic pressure on the PARAM,Q,4.00747 entry. The ElGR entry selects the eigenvalue method for the vibration modes to be calculated and subsequently used for the modal formulation. The rigid body plunge mode and all 10 elastic modes are calculated. They are determined using the Modified Givens (MGIV) method and are normalized to their maximum displacements with MAX.

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The determination of the root bending moment at the side of the fuselage, GRID 11, is the primary objective of this gust analysis. The stress data recovery method used to determine the bending moment is the default Matrix method. (The stress data recovery method is selected by PARAM,DDRMM; the default value is 0 and is more economical when the number of output time steps is large. The Mode Displacement method is selected by PARAM,DDRMM,-1.) To increase the accuracy of the results from the Matrix method, all 11 wing modes are included in the modal formulation. The range of frequencies for which the solution is performed is selected by the FREQ1 40 entry. The frequency range includes the first elastic mode, which is wing bending at f = 2.44 Hz. As more modes are included within the frequency range, the accuracy of the solution increases. Therefore a 10.0 Hz frequency range is chosen that includes the next two elastic modes at 3.56 and 8.53 Hz. 80 frequencies with a Δf = DF = 0.125 Hz interval are selected. Because equally spaced frequencies are selected, the FREQ1 entry is appropriate to input the data. The period of the solution for equally spaced frequencies is T = 1 / Δf = 8.0 sec, i.e., the solution will repeat every 8.0 seconds. The Fourier transformation parameters for SOL 146 must be selected to represent both the resonance characteristics of the structure, the duration of the loading, and the requirement to return to the initial conditions at the end of the period. Closely spaced frequencies are necessary near the resonance frequencies, and small frequency intervals are necessary to obtain a period that is large relative to the duration of the applied loading. For these reasons, the FREQ1 entry is recommended with small, equal frequency intervals. The TSTEP 41 entry defines the output time steps by their number and increment and includes a skip factor for output. The output time steps have no influence on the solution as they do in a transient response analysis by numerical integration; they are only used to specify the output time steps. The period is 8.0 seconds, so 320 time steps of 0.025 seconds increments are specified for printing and plotting. ENDDATA is the last entry in the Bulk Data Section.

Case Control Command The Case Control Section contains the TITLE, SUBTITLE, and LABEL entries. ECHO = BOTH echoes all of the Bulk Data with and without annotations. SPC = 14 selects the constraints. METHOD = 10 selects the ElGR entry for the vibration analysis. SDAMP = 2000 selects the uniform two percent modal structural damping. FREQ = 40 selects the frequency list and TSTEP = 41 selects the time history output. GUST = 1000 selects the gust load. The DLOAD entry is required in SOL 146 in the Case Control Section even when there are only gust loads. SDlSP(PLOT) = 1 and DISP(PLOT) = 2 specify the SET 1 and SET 2 points for determining modal and physical displacements to be plotted. SPCF = 3 causes the constraint forces, i.e., the root bending moment at GRID 11, to be printed. Following OUTPUT(XYOUT) are the XYPLOT requests. Before the first XYPLOT request are several optional statements that define the plot frame and the plot titles. XYPLOTS are requested for the first and second modes and the bending moment at the root. The BEGIN BULK command completes the Case Control Section. The Executive Control Section begins with the identification ID NXN, HA146A that denotes Problem No. HA146A in the Test Problem Library. TIME 10 limits CPU time to 10.0 minutes. SOL 146 calls for the Dynamic Aeroelastic Response Solution. The CEND statement completes the Executive Control Section.

Output The input data are shown in Listing 9-1 and sorted Bulk Data entries are in Listing 9-2. Listing 9-3 presents typical output and is discussed briefly below.

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Under the heading REAL EIGENVALUES are the natural frequencies of the structural model. The first is the rigid body (z-translation) mode followed by 10 elastic modes. Of the 11 modal responses calculated for the aeroelastic response solution, the first and second responses are printed and are plotted in Figure 9-1 and Figure 9-2. The wing aft tip deflection at GRID 10 is also printed and is plotted in Figure 9-3; the limited effect of the wing dynamics is seen as the deflection follows the rigid body mode (see Figure 9-1 and Figure 9-3).

Figure 9-1. Response of Rigid Body Plunge Mode

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Figure 9-2. Response of First Wing Bending Mode

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Figure 9-3. Response of Wing Aft Tip Deflection The root bending moment is printed under the heading of FORCES OF SINGLE-POINT CONSTRAINT for GRID 11 column R3 and is plotted in Figure 9-4. Because certain approximations are made regarding the Fourier series, and due to the method in which the gust load was applied, the printed and plotted output show the absolute response of the SPC forces but only the relative response of the displacements. This is evident for the rigid body plunge mode (Figure 9-1) since this response begins and ends with some nonzero value. Inspection of the plots shows the 0.2 second delay and that the response is damped out by the end of the period. The amplitude of the wing first bending mode in Figure 9-2 goes to zero (the relative displacement returns to its initial constant value) after a few seconds and the wing returns to its trimmed condition. Example 10-5 from Bisplinghoff, Ashley, and Halfman (1955, pp. 682-685) shows the dynamic bending moment calculated at the wing root of the BAH wing at 475 mph due to a sharp-edged gust, including the plunging of the airplane and the first bending mode of the wing, and calculating the aerodynamic loads by quasi-steady aerodynamic strip theory. The comparison

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can therefore only be qualitative because of the many more vibration modes and unsteady aerodynamic lifting surface theory used here. The GENEL entry is used to input the flexibility of the structure and defines nonstandard rotational degrees of freedom for GRID 11. R1 and R2 are rolling and pitching degrees of freedom respectively, and R3 represents the wing root bending degree of freedom. The GENEL for the BAH wing has been discussed in Example HA144B . The column in the output FORCES OF SINGLE-POINT CONSTRAINT labeled R3 provides the desired data for wing root bending moment. The peak response of the wing occurs at about 25 semichords into the gust. The inset in Figure 9-4 is Figure 10-24 of Bisplinghoff, Ashley, and Halfman (1955), and the qualitative comparison is good. The NX Nastran peak response is 2.76 × 108 in-lb (1.9 × 107 ft-lb) in the travel distance of 27 semichords as compared to 2.04 × 108 in-lb (1.7 × 107 ft-lb) in 23 semichords in Bisplinghoff, Ashley, and Halfman (1955).

Figure 9-4. Wing Root Bending Moment[Compares results from Bisplinghoff, Ashley and Halfman (1955, Figure 10-24)]

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N A S T R A N

E X E C U T I V E

C O N T R O L

D E C K

E C H O

ID NXN, HA146A $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA146A $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING FREE IN PLUNGE. $ $ SYMMETRIC RESPONSE TO A SQUARE EDGE GUST . $ $ $ $ SOLUTION TRANSIENT ANALYSIS USING DOUBLET-LATTICE $ $ METHOD AERODYNAMICS AT MACH NO. OF 0.62. $ $ AT MACH NO. OF 0.62 $ $ $ $ OUTPUT XY PLOTS OF MODAL AND GRID POINT $ $ DISPLACEMENTS AND WING ROOT BENDING $ $ MOMENT TIME HISTORIES. $ $ $ $$$$$$$$ $$$$$$$$ TIME 10 $ CPU TIME IN MINUTES SOL 146 $ AEROELASTIC DYNAMIC RESPONSE CEND EXAMPLE HA146A: BAH WING DYNAMIC TRANSIENT ANALYSIS PAGE SYMMETRIC RESPONSE, DOUBLET-LATTICE AERO, 24 BOXES SQUARE-EDGE GUST, ELEVEN MODES C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = EXAMPLE HA146A: BAH WING DYNAMIC TRANSIENT ANALYSIS 2 SUBTITLE = SYMMETRIC RESPONSE, DOUBLET-LATTICE AERO, 24 BOXES 3 LABEL = SQUARE-EDGED GUST, ELEVEN MODES 4 ECHO = BOTH 5 SPC = 14 $ BOUNDARY CONDITIONS (SYMMETRIC, NO PITCH) 6 METHOD = 10 $ MODIFIED GIVENS EIGENVALUE METHOD 7 SDAMP = 2000 $ STRUCTURAL DAMPING (2 PERCENT) 8 GUST = 1000 $ AERODYNAMIC LOADING (SQUARE GUST) 9 DLOAD = 1001 $ REQUIRED 10 FREQ = 40 $ FREQUENCY LIST 11 TSTEP = 41 $ SOLUTION TIME STEPS (1 PERIOD) 12 OUTPUT 13 SET 1 = 1, 2 14 SET 2 = 10 15 SET 3 = 11 16 SDISP(PRINT,PLOT) = 1 $ MODAL DISPLACEMENTS 17 DISP(PRINT,PLOT) = 2 $ STRUCTURAL GRID POINT DISPLACEMENTS 18 SPCF = 3 $ CONSTRAINT FORCES 19 OUTPUT(XYOUT) $ XY PLOTTING PACKAGE 20 CSCALE 2.0 21 PLOTTER = NASTRAN 22 XMIN = 0. 23 XMAX = 8.0 24 UPPER TICS = -1 25 RIGHT TICS = -1 26 XGRID LINES = YES 27 YGRID LINES = YES 28 TCURVE = RIGID BODY PLUNGE MODE 29 XTITLE = TIME(SECONDS) - 1 PERIOD 30 YTITLE = FIRST MODE 31 XYPLOT SDISP / 1(T1) 32 TCURVE = WING BENDING MODE 33 YTITLE = SECOND MODE 34 XYPLOT SDISP / 2(T1) 35 TCURVE = WING TIP DEFLECTION 36 YTITLE = DEFLECTION OF GRID 10 T3 37 XYPLOT DISP/ 10(T3) 38 TCURVE = WING ROOT BENDING MOMENT 39 YTITLE = SPCFORCE OF GRID 11 R3 40 XYPLOT SPCF / 11(R3) 41 BEGIN BULK EXAMPLE HA146A: BAH WING DYNAMIC TRANSIENT ANALYSIS PAGE SYMMETRIC RESPONSE, DOUBLET-LATTICE AERO, 24 BOXES SQUARE-EDGE GUST, ELEVEN MODES I N P U T B U L K D A T A D E C K E C H O

2

3

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * INCLUDE BAH_STRUCT.DAT $ INCLUDE BAH_MASS.DAT $ $ TABDMP1 DEFINES DAMPING COEFFICIENTS (G1) VERSES FREQUENCY (F1). $ (2.0 PERCENT STRUCTURAL DAMPING IS SPECIFIED OVER THE ENTIRE $ FREQUENCY RANGE. VALUES ARE LINEARLY INTERPOLATED AND $ EXTRAPOLATED. A 2.0 PERCENT STRUCTURAL DAMPING IS ROUGHLY $ EQUIVALENT TO 1.0 PERCENT CRITICAL DAMPING.) $ $ ID TYPE TABDMP1 2000 +TABDMP $ F1 G1 F2 G2 "ENDT" +TABDMP 0. .02 10. .02 ENDT

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$ $ * * CONSTRAINTS * * $ $ THE SPC ENTRY CONSTRAINS DOFS. $ $ SID G C D SPC 14 11 45 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * * AERODYNAMIC DATA * * * $ $ (THIS MODEL USES THE LB-IN-SEC SYSTEM) $ $ THE AERO ENTRY DEFINES BASIC AERODYNAMIC PARAMETERS. ACSID IS THE $ AERO COORDINATE SYSTEM. VELOCITY. REFC IS THE REFERENECE $ COORDINATE SYSTEM. RHOREF IS REFERENCE DENSITY. SYMXZ AND $ SYMXY ARE SYMMETRY KEYS. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 8360. 131.25 1.1468-71 $ INCLUDE BAH_AERO5.DAT $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * DYNAMIC LOAD AND RESPONSE DATA * * $ $ GUST DEFINES A STATIONARY VERTICAL GUST. LISTED ARE T/RLOAD ENTRY $ ID, GUST ANGLE OF ATTACK (I.E., THE RATIO OF GUST VELOCITY TO THE $ VEHICLE VELOCITY), LOCATION OF THE GUST WITH RESPECT TO THE ORIGIN $ OF THE AERO COORDINATE SYSTEM, AND THE VEHICLE VELOCITY. $ $ SID DLOAD WG X0 V GUST 1000 1001 1.0 -1672. 8360. $ $ TLOAD1 DEFINES A TIME DEPENDENT DYNAMIC LOAD OR ENFORCED MOTION. $ LISTED ARE THE ID, DAREA ID, DELAY ID, TYPE OF DYNAMIC EXCITATION, $ AND TABELDI ID. $ $ SID DAREA DELAY TYPE TID TLOAD1 1001 1002 1003 $ $ DAREA DEFINES THE DOF WHERE THE LOAD IS APPLIED AND A SCALE FACTOR. $ $ SID P C A DAREA 1002 11 1 0. $ $ TABLED1 DEFINES A TABULAR FUNCTION OF A TIME-DEPENDENT LOAD. $ $ SID TABLED1 1003 +TAB1 $ X1 Y1 X2 Y2 X3 Y3 X4 Y4 +TAB1 0. 1. 1. 1. 1. -1. 2. -1. +TAB2 $ X5 Y5 +TAB2 ENDT $ $ THE MKAERO1 ENTRY DEFINES COMBINATIONS OF MACH NUMBER AND REDUCED $ FREQUENCY EACH OF WHICH WILL BE USED TO GENERATE A MATRIX OF $ GENERALIZED AERODYNAMIC FORCES. $ $ M1 M2 M3 M4 MKAERO1 .62 +MK $ K1 K2 K3 K4 +MK .001 .02 0.10 0.50 $ $ PARAM,GUSTAERO,-1 IS REQUIRED IF GUST LOADS ARE TO BE COMPUTED. $ PARAM GUSTAERO -1 $ $ PARAM,MACH SPECIFIES DYNAMIC PRESSURE. $ PARAM MACH 0.62 $ $ PARAM,Q SPECIFIES DYNAMIC PRESSURE. $ PARAM Q 4.00747 $ $ EIGR DEFINES EIGENVALUE EXTRACTION DATA. LISTED ARE THE $ EIGENVALUE EXTRACTION METHOD, FREQUENCY RANGE, NUMBER OF $ EXPECTED AND DESIRED ROOTS AND THE METHOD OF NORMALIZATION. $ $ SID METHOD F1 F2 NE ND EIGR 10 MGIV 11 +EIGR $ NORM +EIGR MAX $ $ PARAM,LMODES,N SPECIFIES THAT N MODES ARE TO BE USED IN $ THE ANALYSIS. $ PARAM LMODES 11

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$ $ FREQ1 DEFINES THE SET OF FREQUENCIES USED TO OBTAIN $ THE FREQUENCY RESPONSE SOLUTION. LISTED ARE THE STARTING $ FREQUENCY, FREQUENCY INCREMENT AND NUMBER OF INCREMENTS. $ $ SID F1 DF NDF FREQ1 40 0. .125 80 $ $ TSTEP DEFINES TIME STEP INTERVALS AT WHICH THE TRANSIENT $ RESPONSES ARE DESIRED. LISTED ARE THE NUMBER OF STEPS, $ THE TIME INTERVAL AND SKIP FACTOR FOR OUTPUT. $ $ SID N DT NO TSTEP 41 320 .025 1 $ ENDDATA INPUT BULK DATA CARD COUNT = 334

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Listing 9-1. Input Files for Discrete Gust Response of BAH Wing EXAMPLE HA146A: BAH WING DYNAMIC TRANSIENT ANALYSIS PAGE 10 SYMMETRIC RESPONSE, DOUBLET-LATTICE AERO, 24 BOXES SQUARE-EDGE GUST, ELEVEN MODES S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 77 .0 .09 .276 .454 .636 .826 1.0 2AERO 1 8360. 131.25 1.1468-71 3CAERO1 1001 1000 0 4 77 1 +CA1 4+CA1 78.75 0.0 0.0 225.0 35.0 500.0 0.0 100.0 5CMASS2 121 5248.7 1 3 6CMASS2 122 134.9 1 3 2 3 7CMASS2 123 790.3 2 3 8CMASS2 341 9727. 3 3 9CMASS2 342 11005. 3 3 4 3 10CMASS2 343 473. 4 3 11CMASS2 561 3253.6 5 3 12CMASS2 562 -139.7 5 3 6 3 13CMASS2 563 946.3 6 3 14CMASS2 781 2617.8 7 3 15CMASS2 782 21. 7 3 8 3 16CMASS2 783 782.3 8 3 17CMASS2 9101 494.8 9 3 18CMASS2 9102 -7.3 9 3 10 3 19CMASS2 9103 185.2 10 3 20CONM1 1 11 +51 21+51 17400. 4.37+7 +52 22+52 4.35+09 23CORD2R 1 0. 0. 0. 0. 0. -1. +C1 24+C1 -1. 0. 0. 25DAREA 1002 11 1 0. 26EIGR 10 MGIV 11 +EIGR 27+EIGR MAX 28FREQ1 40 0. .125 80 29GENEL 432 1 3 2 3 3 3 +01 30+01 4 3 5 3 6 3 7 3 +02 31+02 8 3 9 3 10 3 +03 32+03 UD 11 3 11 4 11 5 +04 33+04 11 6 +05 34+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 35+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 36+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 37+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 38+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 39+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 40+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 41+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 42+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 43+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 44+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 45+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 46+17 413.0 47GRID 1 20.25 90. 12456 48GRID 2 -81. 90. 12456 49GRID 3 17.85 186. 12456 50GRID 4 -71.4 186. 12456 51GRID 5 15.8 268. 12456 52GRID 6 -63.2 268. 12456 53GRID 7 13.3 368. 12456 54GRID 8 -53.2 368. 12456 55GRID 9 11.05 458. 12456 56GRID 10 -44.2 458. 12456 57GRID 11 0.0 0. 126 58GUST 1000 1001 1.0 -1672. 8360. 59MKAERO1 .62 +MK 60+MK .001 .02 0.10 0.50 61PAERO1 1000 62PARAM GRDPNT 11 63PARAM GUSTAERO-1 64PARAM LMODES 11 65PARAM MACH 0.62 66PARAM Q 4.00747 67PARAM WTMASS .0025907 68SET1 14 1 THRU 11 69SPC 14 11 45 70SPLINE2 100 1001 1005 1024 14 0.0 1.0 0 +SP100 71+SP100 -1.0 -1.0 72TABDMP1 2000 +TABDMP 73+TABDMP 0. .02 10. .02 ENDT 74TABLED1 1003 +TAB1 75+TAB1 0. 1. 1. 1. 1. -1. 2. -1. +TAB2 76+TAB2 ENDT 77TLOAD1 1001 1002 1003 78TSTEP 41 48 .025 1 ENDDATA

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TOTAL COUNT=

79

Listing 9-2. Sorted Bulk Data for Discrete Gust Response of BAH Wing EXAMPLE HA146A: BAH WING DYNAMIC TRANSIENT ANALYSIS PAGE 13 SYMMETRIC RESPONSE, DOUBLET-LATTICE AERO, 24 BOXES SQUARE-EDGE GUST, ELEVEN MODES O U T P U T F R O M G R I D P O I N T W E I G H T G E N E R A T O R REFERENCE POINT = 11 M O * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 * 0.000000E+00 0.000000E+00 4.191900E+04 5.128960E+06 -1.642074E+05 0.000000E+00 * 0.000000E+00 0.000000E+00 5.128960E+06 1.350243E+09 -2.381847E+07 0.000000E+00 * 0.000000E+00 0.000000E+00 -1.642074E+05 -2.381847E+07 4.458782E+09 0.000000E+00 * 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 S * 1.000000E+00 0.000000E+00 0.000000E+00 * * 0.000000E+00 1.000000E+00 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * DIRECTION MASS AXIS SYSTEM (S) MASS X-C.G. Y-C.G. Z-C.G. X 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Y 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 Z 4.191900E+04 3.917256E+00 1.223541E+02 0.000000E+00 I(S) * 7.226942E+08 3.727022E+06 0.000000E+00 * * 3.727022E+06 4.458139E+09 0.000000E+00 * * 0.000000E+00 0.000000E+00 0.000000E+00 * I(Q) * 4.458143E+09 * * 7.226906E+08 * * 0.000000E+00 * Q * 9.977437E-04 9.999995E-01 0.000000E+00 * * -9.999995E-01 9.977437E-04 0.000000E+00 * * 0.000000E+00 0.000000E+00 1.000000E+00 * MODE EXTRACTION NO. ORDER 1 1 2 2 3 3 4 4 5 5 6 6 7 9 8 8 9 11 10 10 11 7 POINT-ID = TIME 0.0 2.500000E-02 5.000000E-02 7.500000E-02 1.000000E-01 1.250000E-01 1.500000E-01 1.750000E-01 2.000000E-01 2.250000E-01 2.500000E-01 2.750000E-01 3.000000E-01 3.250000E-01 3.500001E-01 3.750001E-01 4.000001E-01 4.250001E-01 4.500001E-01 4.750001E-01 5.000001E-01 5.250000E-01 5.500000E-01 5.750000E-01 6.000000E-01 6.249999E-01 6.499999E-01 6.749999E-01 6.999999E-01 7.249998E-01 7.499998E-01 7.749998E-01 7.999998E-01 8.249997E-01

TYPE M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M

EIGENVALUE 1.705303E-13 2.344038E+02 5.021460E+02 2.873470E+03 6.346819E+03 8.746056E+03 1.766041E+04 2.401137E+04 4.211877E+04 6.020940E+04 9.492829E+04 1 T1 1.109709E+03 1.109659E+03 1.109616E+03 1.109698E+03 1.109774E+03 1.109665E+03 1.109628E+03 1.109992E+03 1.109896E+03 1.106718E+03 1.097344E+03 1.079960E+03 1.054306E+03 1.020472E+03 9.781787E+02 9.272354E+02 8.680320E+02 8.011689E+02 7.269859E+02 6.457274E+02 5.578156E+02 4.635912E+02 3.630183E+02 2.559458E+02 1.424827E+02 2.289102E+01 -1.026384E+02 -2.337794E+02 -3.698208E+02 -5.099340E+02 -6.535629E+02 -8.003328E+02 -9.498253E+02 -1.101808E+03

R E A L RADIANS 4.129531E-07 1.531025E+01 2.240862E+01 5.360476E+01 7.966693E+01 9.352035E+01 1.328925E+02 1.549560E+02 2.052286E+02 2.453760E+02 3.081043E+02

E I G E N V A L U E S CYCLES GENERALIZED MASS 6.572353E-08 1.085996E+02 2.436702E+00 7.333811E+00 3.566442E+00 4.860579E+01 8.531462E+00 6.036275E+00 1.267939E+01 1.389629E+01 1.488422E+01 3.997501E+00 2.115050E+01 3.884947E+00 2.466202E+01 3.570773E+00 3.266314E+01 3.142323E+00 3.905281E+01 1.016273E+00 4.903633E+01 8.935863E+00

* * * * * *

GENERALIZED STIFFNESS 1.851951E-11 1.719073E+03 2.440721E+04 1.734505E+04 8.819726E+04 3.496237E+04 6.860977E+04 8.573913E+04 1.323508E+05 6.118916E+04 8.482661E+05

D I S P L A C E M E N T V E C T O R (SOLUTION SET) T2 T3 R1 R2

NX Nastran Aeroelastic Analysis User’s Guide

R3

9-13

Dynamic Aeroelastic Response Analysis

Chapter 9

8.499997E-01 8.749997E-01 8.999997E-01 9.249997E-01 9.499996E-01 9.749996E-01 9.999996E-01 1.025000E+00 1.050000E+00 1.075000E+00 1.100000E+00 1.125000E+00 1.149999E+00 1.174999E+00 1.199999E+00 POINT-ID =

M M M M M M M M M M M M M M M 2

TIME TYPE 0.0 2.500000E-02 5.000000E-02 7.500000E-02 1.000000E-01 1.250000E-01 1.500000E-01 1.750000E-01 2.000000E-01 2.250000E-01 2.500000E-01 2.750000E-01 3.000000E-01 3.250000E-01 3.500001E-01 3.750001E-01 4.000001E-01 4.250001E-01 4.500001E-01 4.750001E-01 5.000001E-01 5.250000E-01 5.500000E-01 5.750000E-01 6.000000E-01 6.249999E-01 6.499999E-01 6.749999E-01 6.999999E-01 7.249998E-01 7.499998E-01 7.749998E-01 7.999998E-01 8.249997E-01 8.499997E-01 8.749997E-01 8.999997E-01 9.249997E-01 9.499996E-01 9.749996E-01 9.999996E-01 1.025000E+00 1.050000E+00 1.075000E+00 1.100000E+00 1.125000E+00 1.149999E+00 1.174999E+00 1.199999E+00 POINT-ID =

M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M 10

TIME TYPE 0.0 2.500000E-02 5.000000E-02 7.500000E-02 1.000000E-01 1.250000E-01 1.500000E-01 1.750000E-01 2.000000E-01 2.250000E-01 2.500000E-01 2.750000E-01 3.000000E-01 3.250000E-01 3.500001E-01 3.750001E-01 4.000001E-01 4.250001E-01 4.500001E-01 4.750001E-01 5.000001E-01

G G G G G G G G G G G G G G G G G G G G G

9-14

-1.256537E+03 -1.414464E+03 -1.575782E+03 -1.740556E+03 -1.909026E+03 -2.081293E+03 -2.256867E+03 -2.435018E+03 -2.615432E+03 -2.797921E+03 -2.981789E+03 -3.166401E+03 -3.352308E+03 -3.540389E+03 -3.729090E+03 T1 -1.064374E-01 -3.560842E-01 -4.877727E-01 -4.154845E-02 3.323708E-01 -9.691008E-02 -2.032860E-01 1.275305E+00 6.706439E-01 -1.178898E+01 -4.526402E+01 -1.003895E+02 -1.690780E+02 -2.407879E+02 -3.066593E+02 -3.585891E+02 -3.884948E+02 -3.909465E+02 -3.655805E+02 -3.162392E+02 -2.493268E+02 -1.738578E+02 -1.010919E+02 -4.173491E+01 -3.091099E+00 1.131507E+01 1.337561E+00 -2.899006E+01 -7.137068E+01 -1.158455E+02 -1.537732E+02 -1.785100E+02 -1.854547E+02 -1.736768E+02 -1.469902E+02 -1.118702E+02 -7.460221E+01 -4.099039E+01 -1.669904E+01 -5.077317E+00 -5.044904E+00 -1.303017E+01 -2.611736E+01 -4.123087E+01 -5.318550E+01 -5.796115E+01 -5.749806E+01 -5.560089E+01 -4.667730E+01 T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

D I S P L A C E M E N T T2 T3

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

V E C T O R R1

D I S P L A C E M E N T T3 1.109629E+03 1.109213E+03 1.108986E+03 1.109663E+03 1.110218E+03 1.109507E+03 1.109184E+03 1.111345E+03 1.111682E+03 1.097962E+03 1.058320E+03 9.901852E+02 8.998873E+02 7.962762E+02 6.878726E+02 5.840244E+02 4.936776E+02 4.216771E+02 3.684155E+02 3.325413E+02 3.105166E+02

NX Nastran Aeroelastic Analysis User’s Guide

(SOLUTION SET) R2

V E C T O R R1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R3

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Dynamic Aeroelastic Response Analysis

5.250000E-01 5.500000E-01 5.750000E-01 6.000000E-01 6.249999E-01 6.499999E-01 6.749999E-01 6.999999E-01 7.249998E-01 7.499998E-01 7.749998E-01 7.999998E-01 8.249997E-01 8.499997E-01 8.749997E-01 8.999997E-01 9.249997E-01 9.499996E-01 9.749996E-01 9.999996E-01 1.025000E+00 1.050000E+00 1.075000E+00 1.100000E+00 1.125000E+00 1.149999E+00 1.174999E+00 1.199999E+00 POINT-ID = TIME TYPE 0.0 2.500000E-02 5.000000E-02 7.500000E-02 1.000000E-01 1.250000E-01 1.500000E-01 1.750000E-01 2.000000E-01 2.250000E-01 2.500000E-01 2.750000E-01 3.000000E-01 3.250000E-01 3.500001E-01 3.750001E-01 4.000001E-01 4.250001E-01 4.500001E-01 4.750001E-01 5.000001E-01 5.250000E-01 5.500000E-01 5.750000E-01 6.000000E-01 6.249999E-01 6.499999E-01 6.749999E-01 6.999999E-01 7.249998E-01 7.499998E-01 7.749998E-01 7.999998E-01 8.249997E-01 8.499997E-01 8.749997E-01 8.999997E-01 9.249997E-01 9.499996E-01 9.749996E-01 9.999996E-01 1.025000E+00 1.050000E+00 1.075000E+00 1.100000E+00 1.125000E+00 1.149999E+00 1.174999E+00 1.199999E+00

G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

11 F O R C E S T1 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0 G 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 O F

2.932557E+02 2.672145E+02 2.208370E+02 1.480881E+02 4.576413E+01 -8.817217E+01 -2.509008E+02 -4.323849E+02 -6.199208E+02 -8.038803E+02 -9.777825E+02 -1.136243E+03 -1.276920E+03 -1.403501E+03 -1.523771E+03 -1.645306E+03 -1.774612E+03 -1.917867E+03 -2.078544E+03 -2.254974E+03 -2.442921E+03 -2.639141E+03 -2.839497E+03 -3.036692E+03 -3.225716E+03 -3.410247E+03 -3.596073E+03 -3.776191E+03

S I N G L E - P O I N T T2 T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

C O N S T R A I N T R1 4.578200E+05 2.287512E+05 -1.919589E+05 -4.963040E+03 4.918563E+05 1.308652E+05 -1.207049E+06 -1.527517E+06 3.077393E+06 1.816936E+07 4.922163E+07 9.688455E+07 1.538111E+08 2.090418E+08 2.559364E+08 2.927564E+08 3.155180E+08 3.159732E+08 2.902716E+08 2.455766E+08 1.934463E+08 1.396292E+08 8.577131E+07 3.860412E+07 9.529455E+06 3.721396E+06 1.563199E+07 3.733358E+07 6.573027E+07 9.842066E+07 1.277767E+08 1.446556E+08 1.460436E+08 1.351522E+08 1.157641E+08 9.035278E+07 6.260421E+07 3.777444E+07 2.002437E+07 1.114935E+07 1.106438E+07 1.752344E+07 2.645404E+07 3.475611E+07 4.198576E+07 4.795212E+07 5.043980E+07 4.639060E+07 3.164836E+07

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R2 1.502993E+05 1.726882E+05 1.331359E+05 9.903859E+04 5.519221E+04 -3.697446E+04 -9.517260E+04 -1.660655E+05 -9.880904E+05 -3.961293E+06 -1.005450E+07 -1.861299E+07 -2.739754E+07 -3.381135E+07 -3.607156E+07 -3.360202E+07 -2.702201E+07 -1.808489E+07 -9.277473E+06 -2.963214E+06 -6.313372E+05 -2.685184E+06 -8.487236E+06 -1.637795E+07 -2.398121E+07 -2.901301E+07 -3.006750E+07 -2.682298E+07 -1.989346E+07 -1.076818E+07 -1.638552E+06 5.257225E+06 8.426425E+06 7.418438E+06 2.688167E+06 -4.500588E+06 -1.220302E+07 -1.834084E+07 -2.143002E+07 -2.087231E+07 -1.684888E+07 -1.031460E+07 -2.972595E+06 3.279618E+06 7.114893E+06 7.923195E+06 5.593115E+06 1.023708E+06 -2.909778E+06

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 R3 2.403533E+05 1.929670E+05 7.908797E+04 3.334882E+04 8.505019E+04 8.820973E+04 -3.746379E+05 -1.280005E+06 6.222305E+03 9.963644E+06 3.548669E+07 7.731285E+07 1.278717E+08 1.772224E+08 2.199131E+08 2.539551E+08 2.748986E+08 2.757133E+08 2.539533E+08 2.155412E+08 1.692089E+08 1.200098E+08 7.112842E+07 2.928766E+07 3.332158E+06 -2.917591E+06 6.952157E+06 2.715102E+07 5.397842E+07 8.380470E+07 1.101674E+08 1.260394E+08 1.283763E+08 1.188569E+08 1.009524E+08 7.787553E+07 5.297147E+07 3.033198E+07 1.400410E+07 6.330624E+06 6.784177E+06 1.252859E+07 2.056996E+07 2.912948E+07 3.668042E+07 4.116635E+07 4.180366E+07 3.960902E+07 3.202975E+07

NX Nastran Aeroelastic Analysis User’s Guide

9-15

Chapter 9

Dynamic Aeroelastic Response Analysis

Listing 9-3. Output for Discrete Gust Response of BAH Wing

9.3

Transient Rolling of BAH Wing Due to Aileron (Example HA146B)

This example considers the roll response of the BAH wing due to an enforced rotation of the aileron. The Lagrange Multiplier method is used to enforce the aileron rotation, a technique that is discussed in Diagnostic Messages for Dynamic Analysis in the NX Nastran Basic Dynamic Analysis User’s Guide . Example 8-4 of Bisplinghoff, Ashley, and Halfman (1955, pp. 467-469) shows the calculation of the aileron effectiveness of the BAH wing as a function of Mach number using aerodynamic strip theory. Example HA144B determined the steady roll rate as an antisymmetric static aeroelastic problem. The present example solves for the steady roll rate from the time history of a single degree-of-freedom rolling maneuver by use of the dynamic aeroelastic response technique. The basic model is the same as before: the structural model is contained in the input files BAH_STRUCT.DAT (Listing 6-4), BAH_MASS.DAT (Listing 6-5), and BAH_AILERON.DAT (Listing 6-6), and the aerodynamic model is contained in the data set BAH_AERO58.DAT (Listing 6-7). Only the new or modified data are discussed below. The dynamic response analysis benefits from the inclusion of some structural damping, and a level of g = 0.03 is included in TABDMP1 2000. The fuselage GRID 11 requires additional constraints in plunge and pitch on SPC 13 but is free in roll. The aileron region of the aerodynamic model also requires a spline to connect the trailing edge region boxes on the surface SPLINE1 104 to the aileron GRIDs 8, 10, and 12 on SET1 15. The type of dynamic excitation applied to the aileron is an enforced rotation, which is enforced by the Lagrange Multiplier method. A negative unit aileron rotation δa = −1.0 rad is selected for the enforced rotation on the right wing in order to cause the wing to roll positively, i.e., the right wing tip moves downward. The desired aileron rotation is obtained by applying an unknown hinge moment to the aileron. The hinge moment is introduced as an “extra” DOF, EPOlNT 115 entry in the Bulk Data Section, which augments the stiffness matrix. Displacements proportional to structural loads imposed at extra points are represented by additional terms in the stiffness matrix at the intersections of columns corresponding to the structural grid points and the additional rows corresponding to the extra points. The DMIG ENFORCE entry defines this relationship in the matrix form that includes both EPOlNT 115 and GRID 12, DOF R2. The matrix ENFORCE automatically augments the structural stiffness matrix via the K2PP = ENFORCE Case Control command. The response analysis is done at the aircraft velocity of 475 mph = 8360 in/s but assuming a Mach number of 0.0 and at four reduced frequencies of 0.001, 0.02, 0.10 and 0.50 as listed on the MKAERO1 entry. The reduced frequencies cover the range of frequencies up to the assumed cut-off frequency of 10.0 Hz and includes the first four antisymmetrical vibration modes that are more than sufficient to represent the motion of a rigid body roll. The selection of Mach number 0.0 neglects compressibility effects. The AERO entry lists the velocity at 8360.0 in/s, along with the reference chord, sea-level density, and the flag for antisymmetrical aerodynamic loads. PARAM,Q selects a value of 4.00747 psi for the dynamic pressure, which corresponds to the velocity of 475 mph at sea level. The TLOAD1 1000 entry identifies the scale factor DAREA 1001, the DELAY 1002, the time-dependent moment by the blank TYPE field, and the loading table TID 1003. DAREA 1001 specifies EPOlNT 115 as the point of load application in the Lagrange Multiplier technique, and an aileron rotation scale factor of -1.0. DELAY 1002 defines a time delay of 0.2 seconds in order to observe that the response has damped out at the end of the previous period in the Fourier analysis, and to avoid having to compute the time history for a complete period.

9-16

NX Nastran Aeroelastic Analysis User’s Guide

Dynamic Aeroelastic Response Analysis

As in the discrete gust analysis (see Example HA146A in the previous section), a square-wave time history of the forcing function is input. The rotation is enforced for 1.0 second on the aileron, then reversed for 1.0 second, and then returned to zero for the roll rate to damp out reasonably well before the next period. TABLED1 1003 provides the tabular description of the aileron input beginning with 1.0 rad for the first second and then reversing to -1.0 rad for the next second. (Note that there is no extrapolation of tabulated data in Fourier analysis, so the return of the aileron to 0.0 rad is implied beyond two seconds.) By the Lagrange Multiplier method outlined above, the moment applied at EPOlNT 115 leads to the unit enforced rotation about the aileron hinge line. The EIGR entry specifies the modified Givens eigenvalue method MGIV and requests all 12 modes to be found and normalized on the largest element. PARAM,LMODES,12 uses all 12 modes in the response calculation. The default is to use all of the modes calculated by EIGR, so LMODES is not needed, but it is included here to emphasize the requirement to have the aileron mode present in the analysis because, in this example, the aileron mode has the highest frequency. If the aileron actuator (CELAS2 3) had a lower stiffness, fewer modes could have been used. The FREQ1 40 entry lists the equally spaced frequencies at which the Fourier analysis is performed. If a period of T = 5.0 seconds is assumed to be sufficient for the motion induced by the 2.0 sec duration aileron excitation to damp out, then the frequency interval becomes Δf = DF = 1/T = 0.2 Hz and the number of equally spaced frequencies up to the assumed cutoff at 10.0 Hz becomes NDF = 50. A response record is requested up to the reversal of the aileron input at 1.20 seconds. The Fourier analysis precedes and is independent of the output time history. The purpose of this analysis is to solve for the steady roll rate, which is found by numerical differentiation of the output time history. A sufficiently accurate derivative can be found using a time interval of Δt = 0.02 sec, so TSTEP 41 prescribes N = 60 and DT = 0.02.

Case Control Commands The Case Control Section contains the TITLE, SUBTITLE, LABEL, and ECHO = BOTH. SPC = 13 selects some of the root constraints. MPC = 1 selects the multipoint constraint equation used to represent the control surface relative motion. METHOD = 10 selects the EIGR entry for the vibration analysis and SVEC = ALL requests the printed eigenvectors. K2PP = ENFORCE selects the DMlG entry. SDAMP = 2000 selects g = 0.03 for the modal damping. DLOAD = 1000 selects the hinge moment applied to the EPOlNT 115. FREQ = 40 selects the frequency list. TSTEP = 41 selects the time history output request. DISP = 2 selects SET 2 for printing the physical displacements of GRIDs 11 and 12. Following OUTPUT(XYOUT) is an xy-plot request. Before the plot request are several optional statements that define the plot frame and plot titles. An XYPLOT is requested for the wing (roll) at GRID 11 about the x-axis. The BEGIN BULK command completes the Case Control Section. The Executive Control Section identifies ID NXN, HA146B, which denotes Problem No. HA146B in the Test Problem Library. TIME 10 limits the CPU time to 10.0 minutes. SOL 146 calls for the Dynamic Aeroelastic Response DMAP sequence. The CEND statement completes the Executive Control Section.

Output The input control and data files are shown in Listing 9-4, the sorted Bulk Data entries are in Listing 9-5. Selected output is shown in Listing 9-6 and is discussed briefly below. The first output is the vibration frequency analysis. This is followed by the first three and the last vibration modes. The first mode is rigid body roll with a computed zero frequency. The

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second mode is first antisymmetric torsion and the third mode is first antisymmetric bending. The twelfth mode is the aileron rotation mode with its 60.0 Hz frequency. The third set of output is the time history of the roll maneuver, which is also plotted in Figure 9-5 as represented by the bank angle of GRID 11, DOF R1. The history does not begin at zero because no initial conditions are imposed in the Fourier analysis and the aileron input was selected to damp to zero roll rate rather than zero bank angle. (Note, however, that the bank angle is initially relatively constant at -0.718 rad and shows the maneuver of the previous period to have damped out.) It is the roll rate that is of interest in this analysis rather than the bank angle, and the roll rate can be obtained by numerical differentiation of the bank angle time history. The roll begins at 0.2 seconds due to the delay on the TLOAD1,1000 entry. The steady roll rate (the constant slope of the bank angle versus time curve) is achieved after 1.0 sec in Figure 9-5. Numerical differentiation between t = 1.18 and 1.20 sec gives a roll rate p = 3.30 rad/sec for the unit aileron rotation δa = 1.0 rad. With the velocity of V = 8360 in/s and the wing semispan of b = 500 in, the steady rolling helix angle becomes pb / 2Vδa = 0.197. This agrees closely with the static aeroelastic solution of Example HA144B , which obtained pb/2Vδa = 0.203. Figure 8-18 of Bisplinghoff, Ashley, and Halfman (1955, p. 469) is reproduced in Figure 9-5 to permit a qualitative comparison between the results of the NX Nastran dynamic Fourier transform solution, based on Lifting Surface Theory (DLM) at Mach zero, and the earlier static aeroelastic results based on Strip Theory with a Prandtl-Glauert compressibility correction. The textbook gives a pb/2Vδa value of 0.061, which can be compared with the 0.197 value cited above. The textbook value can be approached by: reducing the effectiveness of the aileron by 61.5% to match the textbook at zero dynamic pressure and by performing the analysis at M = 0.728, . to match the textbook condition. With these changes NX Nastran gives a steady state roll rate of 0.075.

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Figure 9-5. Roll Response of Fuselage Degree of Freedom. Note that the slope of the roll response curve is roll rate, p.[Inset is from Bisplinghoff, Ashley, and Halfman (1955, Figure 8-18]) N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN,HA146B $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ ANTISYMMETRIC RESPONSE TO A $ ENFORCED AILERON DEFLECTION $ $ SOLUTION TRANSIENT ANALYSIS USING $ DOUBLET LATTICE METHOD AERODYNAMICS $ AT MACH NO. OF 0.0 $ $ OUTPUT XY PLOT OF FUSELAGE ROLL $ $$$$$$$$ TIME 10 $ CPU TIME IN MINUTES SOL 146 $ AEROELASTIC RESPONSE CEND BAH WING DYNAMIC TRANSIENT ANALYSIS HA146B ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO ONE CYCLE OF SQUARE WAVE AILERON DEFLECTION C A S E C O N T R O L D E C K E C H O CARD COUNT

$$$$$$$$ $ $ $ $ $ $ $ $ $ $ $ $$$$$$$

PAGE

2

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1 TITLE = BAH WING DYNAMIC TRANSIENT ANALYSIS HA146B 2 SUBTI = ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO 3 LABEL = ONE CYCLE OF SQUARE WAVE AILERON DEFLECTION 4 ECHO = BOTH 5 SPC = 13 $ BOUNDARY CONDITIONS (ANTISYMMETRIC) 6 MPC = 1 $ CONTROL SURFACE RELATIVE MOTION 7 METHOD = 10 $ MODIFIED-GIVENS EIGENVALUE METHOD 8 SVEC = ALL $ PRINT VIBRATION MODES 9 K2PP = ENFORCE $ EPOINT ADDED VIA DMIG 10 SDAMP = 2000 $ STRUCTURAL DAMPING 11 DLOAD = 1000 $ LOADING 12 FREQ = 40 $ FREQUENCY LIST 13 TSTEP = 41 $ SOLUTION TIME STEPS 14 OUTPUT 15 SET 2 = 11 16 DISP = 2 $ STRUCTURAL GRID POINT DISPLACEMENTS 17 OUTPUT(XYOUT) $ XY PLOTTING PACKAGE 18 CSCALE 2.1 19 PLOTTER NASTRAN 20 CURVELINESYMBOL = 0 21 XMIN = 0. 22 XMAX = 2.0 23 XGRID LINES = YES 24 YGRID LINES = YES 25 UPPER TICS = -1 26 RIGHT TICS = -1 27 XTITLE = TIME(SECONDS) 28 TCURVE = FUSELAGE ROLL 29 YTITLE = RESPONSE OF GRID 11 R1 30 XYPLOT DISP / 11(R1) 31 BEGIN BULK BAH WING DYNAMIC TRANSIENT ANALYSIS HA146B PAGE ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO ONE CYCLE OF SQUARE WAVE AILERON DEFLECTION I N P U T B U L K D A T A D E C K E C H O

3

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . INCLUDE BAH_STRUCT.DAT $ INCLUDE BAH_MASS.DAT $ INCLUDE BAH_AILERON $ $ THE EPOINT ENTRY DEFINES AN "EXTRA" POINT IN THE STRUCTURAL MODEL. $ THIS ONE REPRESENTS A LAGRANGE MULTIPLIER FOR THE AILERON HINGE MOMENT. $ EPOINT 115 $ $ $ TABDMP1 DEFINES DAMPING COEFFICIENTS (G1) VERSES FREQUENCY (F1). $ (3 PERCENT STRUCTURAL DAMPING IS SPECIFIED OVER THE ENTIRE FREQUENCY $ RANGE. VALUES ARE LINEARLY INTERPOLATED AND EXTRAPOLATED. A 3 $ PERCENT STRUCTURAL DAMPING IS ROUGHLY EQUIVALENT TO 1.5 PERCENT $ CRITICAL DAMPING). $ $ ID TYPE TABDMP1 2000 +T2000 $ F1 G1 F2 G2 "ENDT" +T2000 0. .03 10. .03 ENDT $ $ * * CONSTRAINTS * * $ $ THE SPC ENTRY CONSTRAINS DOFS. $ $ SID G C D SPC 13 11 35 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * * AERODYNAMIC DATA * * * $ $ (LB-IN-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AERO ENTRY DEFINES BASIC AERODYNAMIC PARAMETERS. ACSID IS THE $ AERO COORDINATE SYSTEM. VELOCITY. REFC IS THE REFERENECE $ COORDINATE SYSTEM. RHOREF IS REFERENCE DENSITY. SYMXZ AND $ SYMXY ARE SYMMETRY KEYS. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 8360. 131.232 1.1468-7-1 $ INCLUDE BAH_AERO58.DAT $ $ PARAM,Q SPECIFIES DYNAMIC PRESSURE. $ PARAM Q 4.00747 $ $ THE SPLINE1 ENTRY SPECIFIES A SURFACE SPLINE FOR INTERPOLATION $ OVER A REGION (BOX1 THRU BOX2) OF A CAERO PANEL AND THE

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$ STRUCTURAL GRIDS LISTED ON A SET1 FLAGGED BY SETG. DZ IS $ A LINEAR ATTACHMENT FLEXIBLITY. $ $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 104 2001 2005 2018 15 $ $ THE SET1 ENTRY DEFINES THE STRUCTURAL GRID POINTS USED FOR $ INTERPOLATION BY THE SPLINE. $ $ SID G1 G2 G3 G4 G5 G6 SET1 15 8 10 12 $ $ THE MKAERO1 ENTRY DEFINES COMBINATIONS OF MACH NUMBER AND REDUCED $ FREQUENCY EACH OF WHICH WILL BE USED TO GENERATE A MATRIX OF $ GENERALIZED AERODYNAMIC FORCES. $ $ M1 M2 M3 M4 ETC MKAERO1 0.0 +MK $ K1 K2 K3 K4 ETC +MK 0.001 0.02 0.10 0.50 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * DYNAMIC LOAD AND RESPONSE DATA * * $ $ TLOAD1 DEFINES A TIME DEPENDENT DYNAMIC LOAD OR ENFORCED MOTION. $ LISTED ARE THE ID, DAREA ID, DELAY ID, TYPE OF DYNAMIC EXCITATION, $ AND TABLEDI ID. $ $ SID DAREA DELAY TYPE TID TLOAD1 1000 1001 1002 1003 $ $ DAREA DEFINES THE DOF WHERE THE LOAD IS APPLIED AND A SCALE FACTOR. $ $ SID P C A DAREA 1001 115 0 -1.0 $ $ TABLED1 DEFINES A TABULAR FUNCTION OF A TIME-DEPENDENT LOAD. $ $ SID TABLED1 1003 +T1003 $ X1 Y1 X2 Y2 ETC. +T1003 0. 1. 1. 1. 1. -1. 2. -1. +T1003A +T1003A ENDT $ $ THE DELAY ENTRY SPECIFIES THE GRID POINT, DISPLACEMENT COMPONENT $ AND THE TIME DELAY OF ONSET OF THE FORCING FUNCTION. $ $ SID P C T DELAY 1002 115 0 .2 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * * USER SUPPLIED DIRECT MATRIX INPUT DATA * * * $ $ TIN=1 SPECIFIES THAT THE ELEMENTS WILL BE REAL AND SINGLE PRE$ CISION, TOUT=0 SPECIFIES THAT THE TYPE OF MATRIX TO BE GENERATED $ WILL BE DETERMINED INTERNALLY, POLAR=0 SPECIFIES THAT THE INPUT $ WILL BE IN ARGAND DIAGRAM FORM (X,IY). $ $ NAME "0" IFO TIN TOUT POLAR NCOL DMIG ENFORCE 0 1 1 0 $ $ G2 AND C2 SPECIFY THAT A PITCHING MOMENT OF 1.0 IN LB BE APPLIED $ AT THE HINGE-LINE OF THE AILERON. SEE P. 3.5-9 OF THE "HANDBOOK $ FOR DYNAMIC ANALYSIS" FOR A DISCUSSION OF LAGRANGE MULTIPLIERS. $ THE MINUS SIGN ARISES FROM THE DEFINITION OF A POSITIVE AILERON $ DEFLECTION, I.E., TRAILING EDGE DOWN ON THE STARBOARD WING. $ $ NAME G1 C1 G2 C2 X21 DMIG ENFORCE 12 5 115 0 1. $ $ NAME G2 C2 X22 G1 C1 X12 DMIG ENFORCE 115 0 12 5 -1. $ $ EIGR DEFINES EIGENVALUE EXTRACTION DATA. LISTED ARE THE $ EIGENVALUE EXTRACTION METHOD, FREQUENCY RANGE, NUMBER OF $ EXPECTED AND DESIRED ROOTS AND THE METHOD OF NORMALIZATION. $ $ SID METHOD F1 F2 NE ND EIGR 10 MGIV 12 +EIGR $ NORM +EIGR MAX $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N MODES ARE TO BE $ USED IN THE DYNAMIC RESPONSE ANALYSIS. $ PARAM LMODES 12 $ $ FREQ1 DEFINES THE SET OF FREQUENCIES USED TO OBTAIN $ THE FREQUENCY RESPONSE SOLUTION. LISTED ARE THE STARTING

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$ FREQUENCY, FREQUENCY INCREMENT AND NUMBER OF INCREMENTS. $ $ SID F1 DF NDF FREQ1 40 0. .2 50 $ $ TSTEP DEFINES TIME STEP INTERVALS AT WHICH THE TRANSIENT $ RESPONSES ARE DESIRED. LISTED ARE THE NUMBER OF STEPS, $ THE TIME INTERVAL AND SKIP FACTOR FOR OUTPUT. $ $ SID N DT NO TSTEP 41 60 .02 1 $ ENDDATA INPUT BULK DATA ENTRY COUNT = 443

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Listing 9-4. Input Files for Transient Aileron Roll BAH WING DYNAMIC TRANSIENT ANALYSIS HA146B PAGE 12 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO ONE CYCLE OF SQUARE WAVE AILERON DEFLECTION S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 1 0. .09 .21 .33 .45 .56 .66 +AE1 2+AE1 .74 3AEFACT 2 .74 .82 .90 .974 4AEFACT 3 .974 1.00 5AEFACT 4 0. .1875 .375 .625 .750 .875 1.00 6AERO 1 8360. 131.232 1.1468-7-1 7CAERO1 1001 1000 0 5 1 1 +CA1 8+CA1 78.75 0. 0. 225. 35. 500. 0. 100. 9CAERO1 2001 1000 0 2 4 1 +CA2 10+CA2 78.75 0. 0. 225. 35. 500. 0. 100. 11CAERO1 3001 1000 0 5 3 1 +CA3 12+CA3 78.75 0. 0. 225. 35. 500. 0. 100. 13CELAS2 3 5142661.12 5 14CMASS2 121 5248.7 1 3 15CMASS2 122 134.9 1 3 2 3 16CMASS2 123 790.3 2 3 17CMASS2 341 9727. 3 3 18CMASS2 342 11005. 3 3 4 3 19CMASS2 343 473. 4 3 20CMASS2 561 3253.6 5 3 21CMASS2 562 -139.7 5 3 6 3 22CMASS2 563 946.3 6 3 23CMASS2 781 2617.8 7 3 24CMASS2 782 21. 7 3 8 3 25CMASS2 783 782.3 8 3 26CMASS2 9101 494.8 9 3 27CMASS2 9102 -7.3 9 3 10 3 28CMASS2 9103 185.2 10 3 29CONM1 1 11 +51 30+51 17400. 4.37+7 +52 31+52 4.35+09 32CONM1 2 12 +AIL1 33+AIL1 0.0 0.0 +AIL2 34+AIL2 13970.5 35CORD2R 1 0. 0. 0. 0. 0. -1. +C1 36+C1 -1. 0. 0. 37DAREA 1001 115 0 -1.0 38DELAY 1002 115 0 .2 39DMIG ENFORCE 0 1 1 0 40DMIG ENFORCE 12 5 115 0 1. 41DMIG ENFORCE 115 0 12 5 -1. 42EIGR 10 MGIV 12 +EIGR 43+EIGR MAX 44EPOINT 115 45FREQ1 40 0. .2 50 46GENEL 432 1 3 2 3 3 3 +01 47+01 4 3 5 3 6 3 7 3 +02 48+02 8 3 9 3 10 3 +03 49+03 UD 11 3 11 4 11 5 +04 50+04 11 6 +05 51+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 52+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 53+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 54+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 55+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 56+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 57+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 58+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 59+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 60+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 61+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 62+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 63+17 413.0 64GRID 1 20.25 90. 12456 65GRID 2 -81. 90. 12456 66GRID 3 17.85 186. 12456 67GRID 4 -71.4 186. 12456 68GRID 5 15.8 268. 12456 69GRID 6 -63.2 268. 12456 70GRID 7 13.3 368. 12456 71GRID 8 -53.2 368. 12456 72GRID 9 11.05 458. 12456 73GRID 10 -44.2 458. 12456 74GRID 11 0.0 0. 126 75GRID 12 -86.45 368. 1246 76MKAERO1 0.0 +MK 77+MK 0.001 0.02 0.10 0.50 78MPC 1 12 3 -1.0 8 3 1.5 +MPC1 79+MPC1 7 3 -0.5 12 5 33.25 80PAERO1 1000 81PARAM GRDPNT 11 82PARAM LMODES 12 83PARAM Q 4.00747 84PARAM WTMASS .0025907 85SET1 14 1 THRU 11 9-23 NX Nastran Aeroelastic Analysis User’s Guide 86SET1 15 8 10 12 87SPC 13 11 35 88SPLINE1 104 2001 2005 2018 15

Dynamic Aeroelastic Response Analysis

Chapter 9

919293949596979899100101-

SPLINE2 +SP2 SPLINE2 +SP3 TABDMP1 +T2000 TABLED1 +T1003 +T1003A TLOAD1 TSTEP ENDDATA TOTAL COUNT=

102 -1.0 103 -1.0 2000 0. 1003 0. ENDT 1000 41

2001 -1.0 3001 -1.0

2001

2016

14

0.

1.

0

+SP2

3001

3005

14

0.

1.

0

+SP3

.03

10.

.03

ENDT

1.

1.

1.

1.

1001 60

1002 .02

1

+T2000 -1.

2.

-1.

+T1003 +T1003A

1003

102

Listing 9-5. Sorted Bulk Data Entries for Transient Aileron Roll BAH WING DYNAMIC TRANSIENT ANALYSIS HA146B ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO ONE CYCLE OF SQUARE WAVE AILERON DEFLECTION R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED NO. ORDER MASS 1 1 -6.892265E-13 8.301967E-07 1.321299E-07 1.667624E+01 2 2 4.945913E+02 2.223941E+01 3.539512E+00 4.167461E+01 3 3 8.508833E+02 2.916990E+01 4.642534E+00 7.082290E+00 4 4 4.434137E+03 6.658932E+01 1.059802E+01 5.831286E+00 5 5 8.732797E+03 9.344943E+01 1.487294E+01 4.022015E+00 6 7 1.501493E+04 1.225354E+02 1.950212E+01 1.220782E+02 7 9 1.766970E+04 1.329274E+02 2.115605E+01 3.908876E+00 8 8 2.409598E+04 1.552288E+02 2.470543E+01 3.522202E+00 9 11 4.211864E+04 2.052283E+02 3.266309E+01 3.142603E+00 10 10 6.020845E+04 2.453741E+02 3.905250E+01 1.016245E+00 11 6 9.304316E+04 3.050298E+02 4.854699E+01 8.852293E+00 12 12 1.420885E+05 3.769462E+02 5.999286E+01 3.619337E+01 EIGENVALUE = -6.892265E-13 CYCLES = 1.321299E-07 R E A L E I G E N V E C T O R N O . POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 11 G 12 G EIGENVALUE = 4.945913E+02 CYCLES = 3.539512E+00 POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 11 G 12 G EIGENVALUE = 8.508833E+02 CYCLES = 4.642534E+00 POINT ID. TYPE T1 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 11 G 12 G EIGENVALUE = 1.420885E+05 CYCLES = 5.999286E+01

T2

T3 1.965065E-01 1.965065E-01 4.061135E-01 4.061135E-01 5.851529E-01 5.851529E-01 8.034934E-01 8.034934E-01 1.000000E+00 1.000000E+00

PAGE

19

GENERALIZED STIFFNESS -1.149371E-11 2.061190E+04 6.026202E+03 2.585672E+04 3.512344E+04 1.832996E+06 6.906864E+04 8.487088E+04 1.323622E+05 6.118652E+04 8.236453E+05 5.142661E+06 1

R1

R2

R3

2.183406E-03 0.0 R E A L T2

E I G E N V E C T O R T3 -1.785428E-01 4.261678E-01 -2.829573E-01 8.086762E-01 -1.618382E-01 8.397149E-01 4.125781E-02 9.189913E-01 2.618429E-01 1.000000E+00

N O .

2

R1

R2

R3

-5.834532E-04 0.0 R E A L T2

E I G E N V E C T O R T3 -2.061499E-01 -2.630347E-01 -2.729964E-01 -3.807691E-01 -5.540806E-02 -1.530691E-01 4.391988E-01 3.594076E-01 1.000000E+00 9.392283E-01

N O .

3

R1

R2

R3

R2

R3

-2.339169E-03 0.0 R E A L

POINT ID. TYPE T1 T2 1 G 2 G 3 G 4 G 5 G 6 G 7 G NX Nastran Aeroelastic Analysis User’s Guide 9-24

E I G E N V E C T O R T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0

N O . R1

12

Dynamic Aeroelastic Response Analysis

8 9 10 11 12 POINT-ID =

G G G G G

0.0 0.0 0.0 0.0 1.000000E+00 11 D I S P L A C E M E N T

TIME TYPE 0.0 2.000000E-02 4.000000E-02 6.000000E-02 8.000000E-02 1.000000E-01 1.200000E-01 1.400000E-01 1.600000E-01 1.800000E-01 2.000000E-01 2.200000E-01 2.400000E-01 2.600000E-01 2.800000E-01 3.000000E-01 3.200000E-01 3.400000E-01 3.600000E-01 3.800001E-01 4.000001E-01 4.200001E-01 4.400001E-01 4.600001E-01 4.800001E-01 5.000001E-01 5.200001E-01 5.400001E-01 5.600001E-01 5.800000E-01 6.000000E-01 6.200000E-01 6.400000E-01 6.600000E-01 6.799999E-01 6.999999E-01 7.199999E-01 7.399999E-01 7.599999E-01 7.799999E-01 7.999998E-01 8.199998E-01 8.399998E-01 8.599998E-01 8.799998E-01 8.999997E-01 9.199997E-01 9.399997E-01 9.599997E-01 9.799997E-01 9.999996E-01 1.020000E+00 1.040000E+00 1.060000E+00 1.080000E+00 1.100000E+00 1.120000E+00 1.140000E+00 1.159999E+00 1.179999E+00 1.199999E+00

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

V E C T O R R1 -7.184407E-01 -7.183016E-01 -7.187638E-01 -7.193694E-01 -7.196903E-01 -7.199214E-01 -7.205878E-01 -7.216198E-01 -7.220712E-01 -7.211820E-01 -7.196965E-01 -7.195845E-01 -7.218282E-01 -7.242953E-01 -7.221243E-01 -7.107134E-01 -6.887337E-01 -6.585693E-01 -6.242192E-01 -5.890207E-01 -5.550677E-01 -5.237980E-01 -4.959850E-01 -4.706734E-01 -4.445662E-01 -4.132899E-01 -3.738612E-01 -3.261797E-01 -2.723936E-01 -2.152457E-01 -1.572284E-01 -1.008229E-01 -4.846819E-02 -1.387435E-03 4.169792E-02 8.425930E-02 1.299640E-01 1.808141E-01 2.370085E-01 2.980180E-01 3.631130E-01 4.307511E-01 4.981631E-01 5.624685E-01 6.225913E-01 6.799052E-01 7.367909E-01 7.947523E-01 8.541449E-01 9.153894E-01 9.795661E-01 1.047222E+00 1.116904E+00 1.185737E+00 1.251923E+00 1.316347E+00 1.381208E+00 1.447247E+00 1.513113E+00 1.577945E+00 1.643943E+00

R2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Listing 9-6. Output for Transient Aileron Roll

9.4

Random Gust Response of BAH Wing (Example HA146C)

This example shows the BAH wing of Example HA146A responding to a random, rather than discrete, gust. Because atmospheric turbulence is a continuous random phenomenon, it is more appropriately analyzed by a statistical approach instead of the deterministic approach of the discrete gust idealization.

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In this example, only the loading is changed from Example HA146A . Therefore, the same structural and aerodynamic models are used with the vehicle again free to translate vertically without pitching. The new loading data entries are discussed below. The purpose of the analysis is to determine the power spectral density (PSD) function of the root bending moment of the BAH wing at 475 mph at sea level, as has been done by Bisplinghoff, Ashley, and Halfman (1955, p. 691) using Strip Theory and the empirical power spectrum of atmospheric turbulence defined by

Equation 9-1. in which V is in ft/s, and in this example V = 475 mph = 696.67 ft/s. This spectrum results in a mean square gust velocity of

Equation 9-2. NX Nastran requires frequencies to be measured in Hertz. It is convenient to factor out the mean square gust velocity, so the spectrum is rewritten [by multiplying Eq. 9-1

]

Equation 9-3. in Eq. 9-3 is the spectrum input into NX Nastran; is input on the The coefficient of RANDPS entry, and 1/V is input on the GUST entry. The spectrum Sa (f) has units of 1/Hz, and its mean square value should be unity although it will be less because of the truncation error in the integral of Eq. 9-3 due to the finite cutoff frequency. Note that in NX Nastran

Equation 9-4. where fco is the cutoff frequency.

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Because this empirical gust spectrum is not a standard form in NX Nastran (the Dryden and von Kármán forms are standard and are specified via the TABRNDG entry), it must be input in the tabular format using TABRND1. This input provides the frequency content of the gust for the random analysis in a format similar to that of the gust profile provided for the gust time history of the discrete gust transient analysis. The GUST 3002 entry specifies some of the gust characteristics. The DLOAD field refers to the RLOAD1 3002 entry, the scale factor (angle of attack) of the gust WG (gust velocity/forward velocity) is set to 1.0/V for an assumed unit gust velocity amplitude, the reference station for the gust X0 does not apply in this frequency response analysis and is selected as 0.0. The velocity is 8360 in/s. The RLOAD1 3002 entry refers to the DAREA 3003 entry that is a required entry although its scale factor is not used in the gust analysis (see Remark 5 on DAREA entry); however, in this case DAREA is used to apply the spectrum to an EPOINT for the special purpose discussed below. The RLOAD1 entry also refers to the TABLED1 3004, which provides the unit gust frequency response up to the cutoff frequency of 10.0 Hz.

Random Response The RANDPS 1031 entry specifies the PSD for the random analysis called out by the command RANDOM = 1031 in the Case Control section. NX Nastran Random Analysis permits subcases for different excited (J) and applied (K) load sets, but for the gust analysis only an autospectral density is available so the J = K = 1 on the RANDPS entry. It selects the TABRDN1 1032 entry that defines the PSD of the source (the gust) with a unit real coefficient = X + iY = 1.0 + i0.0. The PSD of the gust is defined in a tabular format with frequencies ranging from 0.0 to the cutoff at fco = 10.0 Hz. The MKAERO1 entry selects the Mach numbers and reduced frequencies used in generating the aerodynamic matrices. The Mach number is 0.62 and the reduced frequencies, k = 0.001, 0.02, 0.10, and 0.50, are selected to cover the range of vibration frequencies up to 10.0 Hz.. The PARAM,GUSTAERO,-1 entry is required to obtain the aerodynamic loads for the one-dimensional harmonic gust field in addition to the aerodynamic loads caused by the vibrating structure. When there are no gust loads, neither the PARAM,GUSTAERO nor the GUST entry are required. SOL 146 also requires the flight condition data of Mach number and dynamic pressure as parameters: PARAM,MACH,0.62 and PARAM,Q,4.000747. The EIGR entry selects the GlVENS method (GIV) for determining the 11 vibration modes used in the modal formulation. These modes are the same modes selected and discussed in the discrete gust Example HA146A. The BAH wing is free to plunge vertically, and this degree of freedom results in a frequency response to the harmonic gust field that should be zero at zero frequency. This limit cannot be achieved in NX Nastran for numerical reasons, so the lower frequency is selected as 0.001 Hz on the FREQ 40 entry. The remaining frequencies are equally spaced, as is recommended in the Fourier solution, using the FREQ1 40 entry that selects 79 equally spaced frequencies with an interval of Δf = 0.125 Hz from f = 0.125 up to 10.0Hz. . The turbulence PSD selected for the random response analysis is not automatically output by NX Nastran, and a number of additional entries are required. An extra point EPOINT 50 is introduced and incorporated into the stiffness matrix by means of a DMIG entry. The DMIG STIFF entry adds a value of 1.0 to the structural stiffness matrix and represents the stiffness at the extra point. EPOINT 50 is independent of and not connected to the rest of the model. Its ID must be selected less than the lowest numbered CAERO entry, i.e., 1001. A unit load amplitude for all frequencies is applied to EPOINT 50 by means of RLOAD1 3002 and of DAREA 3003, which selects EPOINT 50 as the degree of freedom loaded by the unit loading condition in

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TABLED1 3004. This results in a unit displacement for EPOINT 50. This technique can also be used to output the conventional NX Nastran gust PSDs that may be selected on the TABRNDG entry. The ENDDATA entry completes the Bulk Data Section.

Case Control Commands The Case Control Section contains the TITLE, SUBTITLE, and LABEL commands. ECHO = BOTH echoes back all of the Bulk Data in annotated and sorted formats. SPC = 14 selects the symmetric boundary condition. METHOD = 10 selects the ElGR entry for determining the vibration modes. SDAMP = 2000 selects the modal damping. FREQ = 40 selects the frequency set for which the frequency response calculations are performed. RANDOM = 1031 specifies that this is a random analysis and refers to the RANDPS entry. K2PP = STIFF refers to the DMlG entry with the same name and specifies that the EPOINT 50 augment the stiffness matrix. OUTPUT heads the requests for print and plot data. The SDISP(PLOT) = 1 and DlSP(PLOT) = 1 specifies the displacements to be determined for EPOINT 50; both of these requests are necessary to print and plot the power spectrum of the gust. SPCF(PHASE) = 3, SET 3 = 11 causes the magnitude and phase of the constraint forces to be generated. SUBCASE 1 is the only loading condition and is referenced by the Bulk Data entry RANDPS 1031. The GUST = 3002 command identifies the gust and the DLOAD = 3002 command selects RLOAD1 3002, which applies the unit load to EPOINT 50. Following the OUTPUT(XYOUT) command are the xy-output requests. For infomation on xy-output, see: •

Plotting in the NX Nastran User’s Guide

Preceding the first xy-output request are several optional statements that define the plot frames and titles. Then, xy-plots are requested for the wing root bending moment with the magnitude and phase on the same page by XYPLOT SPCF/11(R3RM,R3IP). The next xy-output request is for the print and plot of the PSD for the wing root bending moment by XYPRINT,XYPLOT SPCF PSDF/11(R3). The last two xy-output requests are for the print and plot of the displacement PSD for EPOINT 50 by XYPRINT,XYPLOT SDISP PSDF/50(T1)and XYPRINT,XYPLOT DISP PSDF/50(T1). Both requests are necessary to obtain the desired PSD check. This generates the the PSD of the atmospheric turbulence as input on the TABRND1 entry. However, this technique leads to a number of user warning messages from the PLOT module indicating that some plots are not generated. These messages can safely be ignored. The BEGIN BULK command completes the Case Control Section. The Executive Control Section begins with the identification ID NXN, HA146C, which denotes Problem No. HA146C in the Test Problem Library. TIME 10 limits the CPU time to 10.0 minutes. SOL 146 calls for the Dynamic Aeroelastic Response DMAP sequence. The CEND statement completes the Executive Control Section.

Output The input data are shown in Listing 9-7 followed by the sorted Bulk Data entries in Listing 9-8. Selected items in the printed output are shown in Listing 9-9. The requested xy-plots are shown in the Figure 9-6 through Figure 9-9. The first output listed is the symmetric vibration analysis. All 11 modes are calculated for formulating the modal matrices. They are the rigid body plunge mode and the 10 elastic modes. Next is the XY-OUTPUT SUMMARY for the input PSDF. The curve ID is 50(3) referring to EPOINT 50, component 3; the component 3 corresponds to the 50(T1) request according to the Item Code in Plotting in the NX Nastran User’s Guide. The summary includes the RMS value of

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0.995306 and the mean frequency of zero crossings N0 = 1.177425 Hz. The RMS value should be unity, as discussed above, but is slightly less because of the 10.0 Hz cutoff frequency. The third output shown is the input power spectrum, again labeled Curve ID = 50, component 3. This is the input on TABRND1 1032 except for the initial interpolated value at f = 0.001 Hz. The input PSDF is also plotted in Figure 9-6.

Figure 9-6. Power Spectral Density Function of Gust [Eq. 10-3] The COMPLEX FORCES OF SINGLE POINT CONSTRAINT output is the frequency response of the constraint forces at GRID 11 due to the gust and is printed next. The constraint forces are presented in polar format as magnitude and phase. GRID 11 degree of freedom R3 is a nonstandard degree of freedom as defined by the GENEL element and represents the wing root bending moment at the side of the fuselage. (Note: GRID 11, R1 is the bending moment at the airplane centerline, and GRID 11, R2 is the root torque.) The magnitude and phase of the wing root bending moment are plotted in the split frames of Figure 9-7.

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Figure 9-7. Magnitude and Phase of Wing Root Bending Moment The next XY-OUTPUT SUMMARY is for Curve ID 11(8), which is the Item Code for degree of freedom R3 (see Plotting in the NX Nastran User’s Guide). The RMS value of the root bending , and the mean frequency of zero moment is 20,501.9 in-lb for the unit gust rms velocity crossings is N0 = 1.620235 Hz. The last printed output is the PSDF of the root bending moment. It is also plotted along with results obtained from a textbook, in Figure 9-8. The frequency response of the mean square root bending moment here is obtained by rerunning this example with the input PSDF in TABRND1 1032 changed to a unit spectrum. Then the output PSDF is the desired frequency response function, i.e., the mean square of the root bending moment and is plotted along with textbook results, in Figure 9-9.

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Figure 9-8. Power Spectral Density Function of Wing Root Bending Moment The earlier solution of Bisplinghoff, Ashley, and Halfman (1955, pp. 693, 694) is plotted in Figure 10-26 of that textbook as the frequency response of the mean square root bending moment (called the admittance function in the text), and in Figure 10-27 of that same text as the power spectrum of the root bending moment. These two textbook plots are scaled and added to Figure 9-9 and Figure 9-8, respectively, in this guide for comparisons to the present solution.

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Figure 9-9. Frequency Response of Root Bending Moment ¯ the ratio of the root The computed results for the two solutions can be compared by calculating A, mean square of the response to the input. The textbook solution for the elastic airplane is

Equation 9-5. ¯ = 20,502 lb-sec. The present solution also obtained the mean while the present solution is A frequency N0 = 1.620 Hz. The ratio of N0 to the airspeed becomes

Equation 9-6. or, in perhaps more meaningful units, 12.28 crossings per mile of travel.

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N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN,HA146C $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ $ USING A STIFF AILERON. SYMMETRIC $ $ RESPONSE TO A RANDOM GUST. $ $ $ $ SOLUTION TRANSIENT ANALYSIS USING THE $ $ DOUBLET-LATTICE METHOD AERODYNAMICS $ $ AT MACH NO. OF 0.0 $ $ $ $ OUTPUT XY PLOTS OF MODAL AND GRID POINT $ $ DISPLACEMENTS $ $ $ $$$$$$$$ $$$$$$$$ TIME 10 $ CPU TIME IN MINUTES SOL 146 $ DYNAMIC AEROELASTIC RESPONSE CEND BAH JET TRANSPORT WING DYNAMIC ANALYSIS HA146C PAGE 2 SYMMETRIC RESPONSE, 5 STRIPS, DOUBLET-LATTICE AERO RESPONSE TO A RANDOM GUST C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = BAH JET TRANSPORT WING DYNAMIC ANALYSIS HA146C 2 SUBTI = SYMMETRIC RESPONSE, 5 STRIPS, DOUBLET-LATTICE AERO 3 LABEL = RESPONSE TO A RANDOM GUST 4 ECHO = BOTH 5 SPC = 14 $ BOUNDARY CONDITIONS (SYMMETRIC, NO PITCH) 6 METHOD = 10 $ GIVENS EIGENVALUE METHOD 7 SDAMP = 2000 $ STRUCTURAL DAMPING (3 PERCENT SPECIFIED) 8 FREQ = 40 $ FREQUENCY LIST 9 RANDOM = 1031 $ EMPIRICAL PSDF TABULATED 10 K2PP = STIFF $ DMIG STIFFNESS 11 OUTPUT 12 SET 1 = 50 13 SET 3 = 11 14 DISP(PLOT) = 1 $ DISP AND SDISP ARE REQUIRED FOR PSD PLOTS OF 15 SDISP(PLOT) = 1 16 SPCF(PHASE) = 3 $ XY PLOTTING PACKAGE 17 SUBCASE 1 18 LABEL = RANDOM GUST ANALYSIS 19 GUST = 3002 $ APPLY THE RANDOM GUST 20 DLOAD = 3002 $ APPLY LOAD TO EPOINT TO GET PSD OF GUST 21 OUTPUT(XYOUT) $ FREQ RESP PACKAGE (COMPLEX NUMBERS) 22 CSCALE 2.0 23 PLOTTER = NASTRAN 24 $ 25 $ *** THIS APPLIES TO SPLIT FRAMES *** 26 $ 27 TRIGHT TICS = -1 28 BRIGHT TICS = -1 29 XTGRID = YES 30 YTGRID = YES 31 XBGRID = YES 32 YBGRID = YES 33 YTTITLE = MAGNITUDE 34 YBTITLE = PHASE 35 $ 36 $ *** THIS SECTION APPLIES TO WHOLE FRAMES *** 37 $ 38 UPPER TICS = -1 39 RIGHT TICS = -1 40 XGRID LINES = YES 41 YGRID LINES = YES 42 YTITLE = RESPONSE 43 $ 44 $ *** THIS SECTION APPLIES TO ALL FRAMES *** 45 $ 46 XTITLE = FREQUENCY (HZ) 47 CURVELINESYMBOL = 6 48 $ 49 $ *** FREQUENCY RESPONSE PLOT REQUESTS *** 50 $ 51 TCURVE = WING ROOT BENDING MOMENT, GRID 11(R3) 52 XYPLOT SPCF / 11(R3RM,R3IP) 53 $ 54 $ *** RANDOM ANALYSIS OUTPUT REQUESTS *** 55 $ 56 TCURVE = POWER SPECTRAL DENSITY FUNCTION 57 YTITLE = WING ROOT BENDING MOMENT PSD, 11(R3) 58 XYPRINT,XYPLOT SPCF PSDF / 11(R3) 59 $ 60 TCURVE = POWER SPECTRAL DENSITY FUNCTION 61 YTITLE = GUST PSD USING EPOINT 50 FOR OUTPUT 62 XYPRINT,XYPLOT SDISP PSDF / 50(T1) 63 XYPRINT,XYPLOT DISP PSDF / 50(T1) 64 $ 65 BEGIN BULK

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BAH JET TRANSPORT WING DYNAMIC ANALYSIS HA146C PAGE 4 SYMMETRIC RESPONSE, 5 STRIPS, DOUBLET-LATTICE AERO RESPONSE TO A RANDOM GUST I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . INCLUDE BAH_STRUCT.DAT $ INCLUDE BAH_MASS.DAT $ $ TABDMP1 DEFINES DAMPING COEFFICIENTS (G1) VERSES FREQUENCY (F1). $ (3 PERCENT STRUCTURAL DAMPING IS SPECIFIED OVER THE ENTIRE FREQUENCY $ RANGE. VALUES ARE LINEARLY INTERPOLATED AND EXTRAPOLATED. A 3% $ STRUCTURAL DAMPING IS ROUGHLY EQUIVALENT TO 1.5% CRITICAL DAMPING.) $ $ ID TYPE TABDMP1 2000 +TABDMP +TABDMP 0. .03 10. .03 ENDT $ $ * * CONSTRAINTS * * $ $ THE SPC ENTRY CONSTRAINS DOFS. $ $ SID G C D SPC 14 11 45 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * DYNAMIC LOAD AND RESPONSE DATA * * $ $ GUST DEFINES A STATIONARY VERTICAL GUST. LISTED ARE T/RLOAD ENTRY $ ID, RATIO OF GUST VEL/VEHICLE VEL, LOCATION OF THE GUST WITH RESPECT $ TO THE ORIGIN OF THE AERO COORDINATE SYSTEM AND VEHICLE VELOCITY. $ $ SID DLOAD WG X0 V GUST 3002 3002 1.1962-4 0. 8360. $ $ RLOAD1 DEFINES A FREQUENCY DEPENDENT DYNAMIC LOAD. LISTED ARE $ THE ID, DAREA ID, DELAY ID, DPHASE ID AND TABLEDI ENTRIES $ $ SID L M N TC TD RLOAD1 3002 3003 3004 $ $ DAREA DEFINES THE DOF WHERE THE LOAD IS APPLIED AND A SCALE FACTOR. $ $ SID P C A DAREA 3003 50 1. $ $ TABLED1 DEFINES A TABULAR FUNCTION OF A FREQUENCY-DEPENDENT LOAD. $ $ SID TABLED1 3004 +T1004 $ X1 Y1 X2 Y2 ETC. +T1004 0. 1. 10. 1. ENDT $ $ RANDPS DEFINES POWER SPECTRAL DENISITY FACTORS FOR RANDOM $ ANALYSIS. LISTED ARE EXCITED AND APPLIED LOAD SET IDS AND $ THEIR SCALE FACTORS. $ $ SID J K X Y TID RANDPS 1031 1 1 1.0 0.0 1032 $ $ TABRND1 IS THE POWER SPECTRAL DENSITY TABLE. $ TABRND1 1032 +001 +001 0.00 2.8708+0 0.25 1.2641+0 0.50 4.7188-1 0.75 2.3080-1+002 +002 1.00 1.3456-1 1.25 8.7595-2 1.50 6.1402-2 1.75 4.5369-2+003 +003 2.00 3.4865-2 2.25 2.7618-2 2.50 2.2412-2 2.75 1.8547-2+004 +004 3.00 1.5601-2 3.25 1.3304-2 3.50 1.1478-2 3.75 1.0004-2+005 +005 4.00 8.7964-3 4.25 7.7947-3 4.50 6.9547-3 4.75 6.2434-3+006 +006 5.00 5.6359-3 5.25 5.1128-3 5.50 4.6593-3 5.75 4.2636-3+007 +007 6.00 3.9162-3 6.25 3.6095-3 6.50 3.3375-3 6.75 3.0951-3+008 +008 7.00 2.8782-3 7.25 2.6833-3 7.50 2.5076-3 7.75 2.3485-3+009 +009 8.00 2.2042-3 8.25 2.0727-3 8.50 1.9526-3 8.75 1.8427-3+010 +010 9.00 1.7418-3 9.25 1.6490-3 9.50 1.5634-3 9.75 1.4843-3+011 +011 10.00 1.4440-3 ENDT $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * * AERODYNAMIC DATA * * * $ $ (THIS MODEL USES THE LB-IN-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AERO ENTRY DEFINES BASIC AERODYNAMIC PARAMETERS. ACSID IS THE $ AERO COORDINATE SYSTEM. VELOCITY. REFC IS THE REFERENECE $ COORDINATE SYSTEM. RHOREF IS REFERENCE DENSITY. SYMXZ AND $ SYMXY ARE SYMMETRY KEYS. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 8360. 131.232 1.1468-71 $ INCLUDE BAH_AERO5.DAT

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$ $ THE MKAERO1 ENTRY DEFINES COMBINATIONS OF MACH NUMBER AND REDUCED $ FREQUENCY EACH OF WHICH WILL BE USED TO GENERATE A MATRIX OF $ GENERALIZED AERODYNAMIC FORCES. $ $ M1 M2 M3 M4 ETC MKAERO1 .62 +MK $ K1 K2 K3 K4 ETC +MK 0.001 0.02 0.10 0.50 $ $ PARAM,GUSTAERO,-1 IS REQUIRED IF GUST LOADS ARE TO BE COMPUTED. $ PARAM GUSTAERO -1 $ $ PARAM,M SPECIFIES MACH NUMBER. $ PARAM M 0.62 $ $ PARAM,Q SPECIFIES DYNAMIC PRESSURE. $ PARAM Q 4.00747 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ * * VIBRATION AND FREQUENCY RESPONSE DATA * * $ $ EIGR DEFINES EIGENVALUE EXTRACTION DATA. LISTED ARE THE $ EIGENVALUE EXTRACTION METHOD, FREQUENCY RANGE, NUMBER OF $ EXPECTED AND DESIRED ROOTS AND THE METHOD OF NORMALIZATION. $ $ SID METHOD F1 F2 NE ND EIGR 10 GIV 0. 1. 11 +EIGR +EIGR MAX $ $ FREQ AND FREQ1 DEFINE THE SET OF FREQUENCIES USED TO OBTAIN $ THE FREQUENCY RESPONSE SOLUTION. FREQ LISTS ARBITRARILY $ SPACED FREQUENCIES. FREQ1 PROVIDES EQUALLY SPACED FREQUENCIES $ BY SPECIFYING THE STARTING FREQUENCY, THE FREQUENCY INCREMENT $ AND NUMBER OF INCREMENTS. $ $ SID F1 F2 F3 ETC FREQ 40 0.001 $ $ SID F1 DF NDF FREQ1 40 .125 .125 79 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ THE EPOINT ENTRY DEFINES AN "EXTRA" POINT IN THE MODEL. $ EPOINT 50 $ $ DISJOINT DOF ADDED TO THE STRUCTURAL STIFFNESS MATRIX $ WHICH IS USED TO CHECK THE INPUT GUST PSD. $ $ NAME "0" IFO TIN TOUT POLAR NCOL DMIG STIFF 0 6 1 0 DMIG STIFF 50 0 50 0 1. $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ENDDATA INPUT BULK DATA CARD COUNT = 360

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Chapter 9

Dynamic Aeroelastic Response Analysis

Listing 9-7. Input Files for Random Gust Response BAH JET TRANSPORT WING DYNAMIC ANALYSIS HA146C PAGE 12 SYMMETRIC RESPONSE, 5 STRIPS, DOUBLET-LATTICE AERO RESPONSE TO A RANDOM GUST S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AEFACT 77 .0 .09 .276 .454 .636 .826 1.0 2AERO 1 8360. 131.232 1.1468-71 3CAERO1 1001 1000 0 4 77 1 +CA1 4+CA1 78.75 0.0 0.0 225.0 35.0 500.0 0.0 100.0 5CMASS2 121 5248.7 1 3 6CMASS2 122 134.9 1 3 2 3 7CMASS2 123 790.3 2 3 8CMASS2 341 9727. 3 3 9CMASS2 342 11005. 3 3 4 3 10CMASS2 343 473. 4 3 11CMASS2 561 3253.6 5 3 12CMASS2 562 -139.7 5 3 6 3 13CMASS2 563 946.3 6 3 14CMASS2 781 2617.8 7 3 15CMASS2 782 21. 7 3 8 3 16CMASS2 783 782.3 8 3 17CMASS2 9101 494.8 9 3 18CMASS2 9102 -7.3 9 3 10 3 19CMASS2 9103 185.2 10 3 20CONM1 1 11 +51 21+51 17400. 4.37+7 +52 22+52 4.35+09 23CORD2R 1 0. 0. 0. 0. 0. -1. +C1 24+C1 -1. 0. 0. 25DAREA 3003 50 1. 26DMIG STIFF 0 6 1 0 27DMIG STIFF 50 0 50 0 1. 28EIGR 10 GIV 0. 1. 11 +EIGR 29+EIGR MAX 30EPOINT 50 31FREQ 40 0.001 32FREQ1 40 .125 .125 79 33GENEL 432 1 3 2 3 3 3 +01 34+01 4 3 5 3 6 3 7 3 +02 35+02 8 3 9 3 10 3 +03 36+03 UD 11 3 11 4 11 5 +04 37+04 11 6 +05 38+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 39+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 40+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 41+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 42+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 43+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 44+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 45+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 46+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 47+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 48+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 49+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 50+17 413.0 51GRID 1 20.25 90. 12456 52GRID 2 -81. 90. 12456 53GRID 3 17.85 186. 12456 54GRID 4 -71.4 186. 12456 55GRID 5 15.8 268. 12456 56GRID 6 -63.2 268. 12456 57GRID 7 13.3 368. 12456 58GRID 8 -53.2 368. 12456 59GRID 9 11.05 458. 12456 60GRID 10 -44.2 458. 12456 61GRID 11 0.0 0. 126 62GUST 3002 3002 1.1962-40. 8360. 63MKAERO1 .62 +MK 64+MK 0.001 0.02 0.10 0.50 65PAERO1 1000 66PARAM GRDPNT 11 67PARAM GUSTAERO-1 68PARAM M 0.62 69PARAM Q 4.00747 70PARAM WTMASS .0025907 71RANDPS 1031 1 1 1.0 0.0 1032 72RLOAD1 3002 3003 3004 73SET1 14 1 THRU 11 74SPC 14 11 45 75SPLINE2 100 1001 1005 1024 14 0.0 1.0 0 +SP100 76+SP100 -1.0 -1.0 77TABDMP1 2000 +TABDMP 78+TABDMP 0. .03 10. .03 ENDT 79TABLED1 3004 +T1004 80+T1004 0. 1. 10. 1. ENDT 81TABRND1 1032 +001 82+001 0.00 2.8708+00.25 1.2641+00.50 4.7188-10.75 2.3080-1+002 83+002 1.00 1.3456-11.25 8.7595-21.50 6.1402-21.75 4.5369-2+003 84+003 2.00 3.4865-22.25 2.7618-22.50 2.2412-22.75 1.8547-2+004 85+004 3.00 1.5601-23.25 1.3304-23.50 1.1478-23.75 1.0004-2+005 NX Nastran Aeroelastic Analysis User’s Guide 9-36 86+005 4.00 8.7964-34.25 7.7947-34.50 6.9547-34.75 6.2434-3+006 87+006 5.00 5.6359-35.25 5.1128-35.50 4.6593-35.75 4.2636-3+007 88+007 6.00 3.9162-36.25 3.6095-36.50 3.3375-36.75 3.0951-3+008

Dynamic Aeroelastic Response Analysis

92-

+011 10.00 1.4440-3ENDT ENDDATA TOTAL COUNT= 93

Listing 9-8. Sorted Bulk Data Entries for Random Gust Response

BAH JET TRANSPORT WING DYNAMIC ANALYSIS SYMMETRIC RESPONSE, 5 STRIPS, DOUBLET-LATTICE RESPONSE TO A RANDOM GUST R E A MODE EXTRACTION EIGENVALUE RADIANS NO. ORDER 1 8 1.275478E-12 1.129371E-06 2 9 2.344038E+02 1.531025E+01 3 10 5.021460E+02 2.240862E+01 4 7 2.873470E+03 5.360476E+01 5 6 6.346819E+03 7.966693E+01 6 11 8.746056E+03 9.352035E+01 7 5 1.766041E+04 1.328925E+02 8 4 2.401137E+04 1.549560E+02 9 3 4.211877E+04 2.052286E+02 10 2 6.020940E+04 2.453760E+02 11 1 9.492829E+04 3.081043E+02

HA146C

PAGE

18

AERO L E I G E N V A L U E S CYCLES GENERALIZED MASS 1.797449E-07 1.085996E+02 2.436702E+00 7.333811E+00 3.566442E+00 4.860579E+01 8.531462E+00 6.036275E+00 1.267939E+01 1.389629E+01 1.488422E+01 3.997501E+00 2.115050E+01 3.884947E+00 2.466202E+01 3.570773E+00 3.266314E+01 3.142323E+00 3.905281E+01 1.016273E+00 4.903633E+01 8.935863E+00

GENERALIZED STIFFNESS 1.385164E-10 1.719073E+03 2.440721E+04 1.734505E+04 8.819726E+04 3.496237E+04 6.860977E+04 8.573913E+04 1.323508E+05 6.118916E+04 8.482661E+05

X Y - O U T P U T S U M M A R Y ( A U T O O R P S D F ) PLOT CURVE FRAME RMS NO. POSITIVE XMIN FOR XMAX FOR YMIN FOR X FOR YMAX FOR X FOR* TYPE TYPE NO. CURVE ID. VALUE CROSSINGS ALL DATA ALL DATA ALL DATA YMIN ALL DATA YMAX PSDF DISP 1 50( 3) 9.953058E-01 1.177425E+00 1.000E-03 1.000E+01 1.444E-03 1.000E+01 2.864E+00 1.000E-03 P R I N T E D D A T A F O R T H I S C U R V E F O L L O W S DISPLACEMENT CURVE ID = 50 COMPONENT = 3 WHOLE FRAME PRINT NUMBER X-VALUE Y-VALUE 1 1.000000E-03 2.864373E+00 2 1.250000E-01 2.067450E+00 3 2.500000E-01 1.264100E+00 4 3.750000E-01 8.679900E-01 5 5.000000E-01 4.718800E-01 6 6.250000E-01 3.513400E-01 7 7.500000E-01 2.308000E-01 8 8.749999E-01 1.826800E-01 9 1.000000E+00 1.345600E-01 10 1.125000E+00 1.110775E-01 11 1.250000E+00 8.759500E-02 12 1.375000E+00 7.449849E-02 13 1.500000E+00 6.140199E-02 14 1.625000E+00 5.338550E-02 15 1.750000E+00 4.536900E-02 16 1.875000E+00 4.011700E-02 17 2.000000E+00 3.486500E-02 18 2.125000E+00 3.124150E-02 19 2.250000E+00 2.761800E-02 20 2.375000E+00 2.501500E-02 21 2.500000E+00 2.241200E-02 22 2.625000E+00 2.047950E-02 23 2.750000E+00 1.854700E-02 24 2.875000E+00 1.707400E-02 25 3.000000E+00 1.560100E-02 26 3.125000E+00 1.445250E-02 27 3.250000E+00 1.330400E-02 28 3.375000E+00 1.239100E-02 29 3.500000E+00 1.147800E-02 30 3.625000E+00 1.074100E-02 31 3.750000E+00 1.000400E-02 32 3.875000E+00 9.400200E-03 33 4.000000E+00 8.796400E-03 34 4.125000E+00 8.295550E-03 35 4.250000E+00 7.794700E-03 36 4.375000E+00 7.374700E-03 37 4.500000E+00 6.954700E-03

NX Nastran Aeroelastic Analysis User’s Guide

9-37

Chapter 9

Dynamic Aeroelastic Response Analysis

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 POINT-ID =

FREQUENCY 1.000000E-03

11 C O M P L E X TYPE G

1.250000E-01

G

2.500000E-01

G

3.750000E-01

G

5.000000E-01

G

6.250000E-01

G

7.500000E-01

G

8.749999E-01

G

1.000000E+00

G

1.125000E+00

G

1.250000E+00

G

1.375000E+00

G

1.500000E+00

G

1.625000E+00

G

1.750000E+00

G

1.875000E+00

G

2.000000E+00

G

2.125000E+00

G

2.250000E+00

G

2.375000E+00

G

2.500000E+00

G

9-38

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

F O R C E S T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

4.625000E+00 4.750000E+00 4.875000E+00 5.000000E+00 5.125000E+00 5.250000E+00 5.375000E+00 5.500000E+00 5.625000E+00 5.750000E+00 5.875000E+00 6.000000E+00 6.125000E+00 6.250000E+00 6.375000E+00 6.500000E+00 6.625000E+00 6.750000E+00 6.875000E+00 7.000000E+00 7.125000E+00 7.250000E+00 7.375000E+00 7.500000E+00 7.625000E+00 7.750000E+00 7.875000E+00 8.000000E+00 8.125000E+00 8.249999E+00 8.375000E+00 8.500000E+00 8.625000E+00 8.750000E+00 8.875000E+00 9.000000E+00 9.124999E+00 9.250000E+00 9.375000E+00 9.500000E+00 9.625000E+00 9.749999E+00 9.875000E+00 1.000000E+01

6.599050E-03 6.243400E-03 5.939650E-03 5.635900E-03 5.374350E-03 5.112801E-03 4.886050E-03 4.659301E-03 4.461450E-03 4.263601E-03 4.089901E-03 3.916200E-03 3.762851E-03 3.609500E-03 3.473500E-03 3.337500E-03 3.216300E-03 3.095100E-03 2.986650E-03 2.878200E-03 2.780750E-03 2.683301E-03 2.595450E-03 2.507600E-03 2.428050E-03 2.348500E-03 2.276350E-03 2.204200E-03 2.138450E-03 2.072701E-03 2.012650E-03 1.952600E-03 1.897650E-03 1.842700E-03 1.792250E-03 1.741800E-03 1.695400E-03 1.649000E-03 1.606200E-03 1.563400E-03 1.523850E-03 1.484300E-03 1.464150E-03 1.444000E-03

O F S I N G L E (MAGNITUDE/PHASE) T3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

P O I N T

R1 8.312178E+02 173.6820 1.092055E+04 71.1811 1.861192E+04 50.7622 2.326442E+04 36.7060 2.614285E+04 26.6160 2.814698E+04 18.9874 2.976732E+04 12.8939 3.127258E+04 7.7702 3.282730E+04 3.2554 3.455186E+04 359.0998 3.655425E+04 355.1104 3.894966E+04 351.1161 4.187598E+04 346.9409 4.550782E+04 342.3771 5.006727E+04 337.1540 5.581818E+04 330.8942 6.299762E+04 323.0629 7.155795E+04 312.9433 8.049985E+04 299.7882 8.702245E+04 283.4468 8.756493E+04 265.3443

C O N S T R A I N T

R2 3.395612E+01 359.8732 1.230243E+03 248.5424 2.087676E+03 229.3616 2.585104E+03 215.6992 2.865934E+03 205.8111 3.031500E+03 198.3330 3.135313E+03 192.3960 3.204327E+03 187.4722 3.252165E+03 183.2352 3.285456E+03 179.4765 3.306606E+03 176.0588 3.314806E+03 172.8939 3.305959E+03 169.9370 3.271766E+03 167.2026 3.198116E+03 164.8229 3.063984E+03 163.2185 2.847984E+03 163.5802 2.576946E+03 169.0459 2.521016E+03 184.3782 3.313615E+03 201.3680 5.049220E+03 204.1534

R3 8.308215E+02 174.6252 9.330418E+03 71.8331 1.589639E+04 51.0190 1.987951E+04 36.7986 2.235717E+04 26.6021 2.409714E+04 18.8895 2.551859E+04 12.7225 2.685191E+04 7.5306 2.823920E+04 2.9501 2.978557E+04 358.7296 3.158650E+04 354.6748 3.374499E+04 350.6140 3.638537E+04 346.3704 3.966609E+04 341.7360 4.379009E+04 336.4393 4.900098E+04 330.1027 5.552409E+04 322.1909 6.333834E+04 311.9869 7.157834E+04 298.7429 7.775445E+04 282.3080 7.864403E+04 264.1071

Dynamic Aeroelastic Response Analysis

2.625000E+00

G

2.750000E+00

G

2.875000E+00

G

3.000000E+00

G

3.125000E+00

G

3.250000E+00

G

3.375000E+00

G

3.500000E+00

G

3.625000E+00

G

3.750000E+00

G

3.875000E+00

G

4.000000E+00

G

4.125000E+00

G

4.250000E+00

G

4.375000E+00

G

4.500000E+00

G

4.625000E+00

G

4.750000E+00

G

4.875000E+00

G

5.000000E+00

G

5.125000E+00

G

5.250000E+00

G

5.375000E+00

G

5.500000E+00

G

5.625000E+00

G

5.750000E+00

G

5.875000E+00

G

6.000000E+00

G

6.125000E+00

G

6.250000E+00

G

6.375000E+00

G

6.500000E+00

G

6.625000E+00

G

6.750000E+00

G

6.875000E+00

G

7.000000E+00

G

7.125000E+00

G

7.250000E+00

G

7.375000E+00

G

7.500000E+00

G

7.625000E+00

G

7.750000E+00

G

7.875000E+00

G

8.000000E+00

G

8.125000E+00

G

8.249999E+00

G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

8.187171E+04 248.1337 7.365701E+04 233.4971 6.653845E+04 221.0725 6.234854E+04 208.7704 6.116080E+04 191.8247 5.440966E+04 161.7768 3.006864E+04 132.3735 1.407659E+04 132.2889 8.872051E+03 151.2876 7.594335E+03 168.1369 7.170586E+03 177.4938 6.835414E+03 182.4707 6.472588E+03 185.3352 6.087274E+03 187.1697 5.697985E+03 188.4894 5.318625E+03 189.5563 4.957508E+03 190.5126 4.618969E+03 191.4413 4.304840E+03 192.3934 4.015488E+03 193.4026 3.750462E+03 194.4919 3.508895E+03 195.6777 3.289765E+03 196.9724 3.092037E+03 198.3841 2.914741E+03 199.9180 2.757052E+03 201.5756 2.618288E+03 203.3531 2.497961E+03 205.2426 2.395750E+03 207.2279 2.311544E+03 209.2858 2.245442E+03 211.3835 2.197763E+03 213.4780 2.169088E+03 215.5149 2.160351E+03 217.4304 2.172853E+03 219.1474 2.208555E+03 220.5804 2.270234E+03 221.6316 2.361873E+03 222.1888 2.489371E+03 222.1230 2.661321E+03 221.2744 2.890586E+03 219.4369 3.195997E+03 216.3201 3.605222E+03 211.4979 4.154430E+03 204.3017 4.874108E+03 193.6824 5.720918E+03

7.199502E+03 197.1053 9.551510E+03 187.0322 1.242790E+04 175.9375 1.664170E+04 162.5133 2.349440E+04 141.8488 3.066718E+04 104.3605 2.526187E+04 61.0607 1.658689E+04 36.5247 1.142868E+04 23.9947 8.424339E+03 16.6051 6.527583E+03 11.6843 5.241428E+03 8.1275 4.320244E+03 5.4098 3.632354E+03 3.2509 3.101771E+03 1.4870 2.681875E+03 0.0153 2.342602E+03 358.7673 2.063747E+03 357.6955 1.831257E+03 356.7653 1.635066E+03 355.9508 1.467788E+03 355.2320 1.323882E+03 354.5926 1.199116E+03 354.0196 1.090203E+03 353.5018 9.945497E+02 353.0295 9.100891E+02 352.5945 8.351497E+02 352.1884 7.683656E+02 351.8042 7.086125E+02 351.4343 6.549512E+02 351.0716 6.065951E+02 350.7089 5.628750E+02 350.3384 5.232190E+02 349.9518 4.871305E+02 349.5409 4.541738E+02 349.0944 4.239588E+02 348.6017 3.961263E+02 348.0492 3.703334E+02 347.4201 3.462328E+02 346.6966 3.234458E+02 345.8563 3.015194E+02 344.8768 2.798513E+02 343.7400 2.575742E+02 342.4680 2.334054E+02 341.2171 2.057729E+02 340.6236 1.749926E+02

7.393528E+04 246.7932 6.690588E+04 232.0486 6.081651E+04 219.5118 5.736808E+04 207.0944 5.668552E+04 190.0315 5.084297E+04 159.8614 2.837297E+04 130.3016 1.343349E+04 129.9028 8.537994E+03 148.3866 7.345599E+03 164.8525 6.978353E+03 173.9513 6.702518E+03 178.6795 6.400520E+03 181.2695 6.074047E+03 182.7957 5.739547E+03 183.7707 5.409977E+03 184.4546 5.093353E+03 184.9872 4.794036E+03 185.4486 4.514035E+03 185.8867 4.253931E+03 186.3322 4.013496E+03 186.8046 3.792052E+03 187.3176 3.588719E+03 187.8814 3.402559E+03 188.5032 3.232650E+03 189.1890 3.078166E+03 189.9436 2.938391E+03 190.7699 2.812764E+03 191.6710 2.700880E+03 192.6471 2.602533E+03 193.6978 2.517735E+03 194.8195 2.446741E+03 196.0057 2.390105E+03 197.2444 2.348757E+03 198.5189 2.324048E+03 199.8016 2.317954E+03 201.0558 2.333218E+03 202.2285 2.373637E+03 203.2455 2.444575E+03 204.0064 2.553544E+03 204.3697 2.711320E+03 204.1369 2.933142E+03 203.0152 3.240458E+03 200.5678 3.659708E+03 196.1075 4.208632E+03 188.5583 4.837537E+03

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0.0 0.0 0.0 178.2120 342.8162 176.4560 0.0 0.0 0.0 6.403101E+03 1.522189E+02 5.290097E+03 0.0 0.0 0.0 156.9885 351.7622 158.8477 8.500000E+00 G 0.0 0.0 0.0 6.401133E+03 1.580326E+02 5.146721E+03 0.0 0.0 0.0 132.2299 3.0602 137.8788 8.625000E+00 G 0.0 0.0 0.0 5.689188E+03 1.791391E+02 4.426384E+03 0.0 0.0 0.0 109.1909 5.9914 118.7027 8.750000E+00 G 0.0 0.0 0.0 4.799419E+03 1.928594E+02 3.587265E+03 0.0 0.0 0.0 90.7374 3.7587 104.0497 8.875000E+00 G 0.0 0.0 0.0 4.061349E+03 1.976025E+02 2.892257E+03 0.0 0.0 0.0 76.3440 0.5925 93.2252 9.000000E+00 G 0.0 0.0 0.0 3.525807E+03 1.973737E+02 2.372418E+03 0.0 0.0 0.0 64.7253 357.6738 84.7535 9.124999E+00 G 0.0 0.0 0.0 3.156222E+03 1.949030E+02 1.992135E+03 0.0 0.0 0.0 54.9671 355.1408 77.5275 9.250000E+00 G 0.0 0.0 0.0 2.909021E+03 1.915779E+02 1.713875E+03 0.0 0.0 0.0 46.5406 352.9121 70.8558 9.375000E+00 G 0.0 0.0 0.0 2.750960E+03 1.880790E+02 1.510536E+03 0.0 0.0 0.0 39.1600 350.8983 64.3460 9.500000E+00 G 0.0 0.0 0.0 2.658426E+03 1.847394E+02 1.363717E+03 0.0 0.0 0.0 32.6488 349.0218 57.7964 9.625000E+00 G 0.0 0.0 0.0 2.615604E+03 1.817386E+02 1.261552E+03 0.0 0.0 0.0 26.9021 347.2419 51.1629 9.749999E+00 G 0.0 0.0 0.0 2.611242E+03 1.791593E+02 1.195620E+03 0.0 0.0 0.0 21.8159 345.5139 44.4862 9.875000E+00 G 0.0 0.0 0.0 2.637926E+03 1.770507E+02 1.160150E+03 0.0 0.0 0.0 17.3079 343.8145 37.8807 1.000000E+01 G 0.0 0.0 0.0 2.690929E+03 1.754470E+02 1.151029E+03 0.0 0.0 0.0 13.3131 342.1386 31.4983 X Y - O U T P U T S U M M A R Y ( A U T O O R P S D F ) PLOT CURVE FRAME RMS NO. POSITIVE XMIN FOR XMAX FOR YMIN FOR X FOR YMAX FOR X FOR* TYPE TYPE NO. CURVE ID. VALUE CROSSINGS ALL DATA ALL DATA ALL DATA YMIN ALL DATA YMAX PSDF SPCF 3 11( 8) 2.050194E+04 1.620235E+00 1.000E-03 1.000E+01 1.913E+03 1.000E+01 3.430E+08 3.750E-01 P R I N T E D D A T A F O R T H I S C U R V E F O L L O W S S P C F CURVE ID = 11 COMPONENT = 8 WHOLE FRAME PRINT NUMBER X-VALUE Y-VALUE 1 1.000000E-03 1.977175E+06 2 1.250000E-01 1.799854E+08 3 2.500000E-01 3.194319E+08 4 3.750000E-01 3.430251E+08 5 5.000000E-01 2.358660E+08 6 6.250000E-01 2.040133E+08 7 7.500000E-01 1.502966E+08 8 8.749999E-01 1.317169E+08 9 1.000000E+00 1.073052E+08 10 1.125000E+00 9.854578E+07 11 1.250000E+00 8.739414E+07 12 1.375000E+00 8.483326E+07 13 1.500000E+00 8.128981E+07 14 1.625000E+00 8.399666E+07 15 1.750000E+00 8.699830E+07 16 1.875000E+00 9.632476E+07 17 2.000000E+00 1.074861E+08 18 2.125000E+00 1.253330E+08 19 2.250000E+00 1.414997E+08 20 2.375000E+00 1.512345E+08 21 2.500000E+00 1.386156E+08 22 2.625000E+00 1.119497E+08 23 2.750000E+00 8.302371E+07 24 2.875000E+00 6.315071E+07 25 3.000000E+00 5.134440E+07 26 3.125000E+00 4.643948E+07 27 3.250000E+00 3.439094E+07 28 3.375000E+00 9.975069E+06 29 3.500000E+00 2.071304E+06 30 3.625000E+00 7.829903E+05 31 3.750000E+00 5.397941E+05 32 3.875000E+00 4.577653E+05 33 4.000000E+00 3.951672E+05 34 4.125000E+00 3.398409E+05 35 4.250000E+00 2.875781E+05 36 4.375000E+00 2.429403E+05 37 4.500000E+00 2.035491E+05 38 4.625000E+00 1.711942E+05 39 4.750000E+00 1.434907E+05 40 4.875000E+00 1.210293E+05 41 5.000000E+00 1.019869E+05 42 5.125000E+00 8.657085E+04 43 5.250000E+00 7.352034E+04 44 5.375000E+00 6.292698E+04 45 5.500000E+00 5.394262E+04 46 5.625000E+00 4.662227E+04 47 5.750000E+00 4.039806E+04 48 5.875000E+00 3.531279E+04 49 6.000000E+00 3.098357E+04 50 6.125000E+00 2.744907E+04 51 6.250000E+00 2.444779E+04 52 6.375000E+00 2.201848E+04 53 6.500000E+00 1.998008E+04 54 6.625000E+00 1.837344E+04 55 6.750000E+00 1.707461E+04 8.375000E+00

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56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

6.875000E+00 7.000000E+00 7.125000E+00 7.250000E+00 7.375000E+00 7.500000E+00 7.625000E+00 7.750000E+00 7.875000E+00 8.000000E+00 8.125000E+00 8.249999E+00 8.375000E+00 8.500000E+00 8.625000E+00 8.750000E+00 8.875000E+00 9.000000E+00 9.124999E+00 9.250000E+00 9.375000E+00 9.500000E+00 9.625000E+00 9.749999E+00 9.875000E+00 1.000000E+01

1.613149E+04 1.546431E+04 1.513814E+04 1.511812E+04 1.551027E+04 1.635102E+04 1.784922E+04 2.020490E+04 2.390297E+04 2.952186E+04 3.787747E+04 4.850485E+04 5.632427E+04 5.172190E+04 3.718041E+04 2.371273E+04 1.499244E+04 9.803491E+03 6.728371E+03 4.843722E+03 3.664899E+03 2.907492E+03 2.425229E+03 2.121818E+03 1.970670E+03 1.913110E+03

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Listing 9-9. Output for Random Gust Response

9.5 Frequency Response of BAH Wing to Oscillating Aileron (Examples HA146D and HA146DR) Each of the four previous dynamic aeroelastic response examples performs a frequency response analysis. This is followed by either a transformation into the time domain via an inverse Fourier transform or by postprocessing of the data for random analysis. The purpose of this example is to demonstrate a straightforward application of the dynamic aeroelastic response capability, which is entirely in the frequency domain, and to outline the restart capability. Both the enforced displacement of the aileron and the output are examined in the frequency domain. The structural and aerodynamic models are identical to the earlier transient roll model (Example HA146B ) but here the enforced displacement is oscillatory. The Lagrange multiplier information, the EPOlNT and the DMlG, all related to the loading in the transient roll example, are the same. The only difference here is that the frequency-dependent enforced aileron rotation is input via the RLOAD1 entry in contrast to the TLOAD1 entry required previously for the time history of the transient enforced rotation. RLOAD1 1000 specifies DAREA 1001, which provides the scale factor A = −1.0 for the magnitude of the aileron rotation (applied to the EPOINT in the Lagrange multiplier technique), DELAY 1002, which has no effect on the frequency response solution, and TABLED1 1003 for the frequency response of the enforced rotation. The frequency response solution is performed over 50 equally spaced intervals with an increment of Δf = 0.1 Hz for a total range of 5.0 Hz via the FREQ1 40 entry even though TABLED1 covers an input frequency range up to 10.0 Hz. The 5.0 Hz range includes the first three modes determined by the vibration analysis. The first mode is the rigid body roll mode, the second mode is first wing torsion, and third mode is the wing first antisymmetric bending mode. The ENDDATA entry completes the Bulk Data Section.

Case Control Command The Case Control Section is similar to that for the transient roll example. The K2PP request for the Lagrange multiplier is still required but the TSTEP output request has been deleted and the displacement output request has been replaced with a constraint force output request: SPCF = 1, where SET 1 is the fuselage GRID 11. The plot request is different from the plot request in HA146B. Here the frequency responses of wing root bending moment and torque are requested in polar form; split frames present the magnitude above the phase angle as functions of the frequency. The constraint force at GRID 11, DOF R3 is the bending moment at the side of the fuselage, and the force at GRID 11, DOF R2 is the torque. The BEGIN BULK entry completes the Case Control Section. The Executive Control Section identifies the problem as ID NXN, HA146D, which denotes Problem No. HA146D in the Test Problem Library. The time is limited to 10.0 CPU minutes and SOL 146 calls for the dynamic aeroelastic response DMAP sequence. The CEND entry completes the Executive Control Section.

Output The input data and the sorted Bulk Data entries are shown in Listing 9-10 and Listing 9-11. Selected outputs are in Listing 9-12, Figure 9-10 and Figure 9-11 and are discussed as follows.

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The antisymmetric frequencies of the BAH wing are shown first in Listing 9-12. The first frequency is a computed zero for the rigid body roll mode. The second frequency is 3.540 Hz for the first torsion mode, and the third frequency is 4.645 Hz for the first antisymmetric bending mode. The frequency responses for the root shear (T3), the root torque (R2), and the root bending moment (R3) are listed next under COMPLEX FORCES OF SINGLE POINT CONSTRAINT. The root bending moment and torque are plotted in Figure 9-10 and Figure 9-11, respectively. The expected resonances in the frequency response are found in the tabulated and plotted data: the root bending moment peaks near the first bending frequency and the root torque peaks near the first torsion frequency.

Figure 9-10. Wing Root Bending Moment

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Figure 9-11. Wing Root Torque This example is very similar to the HA146B example and is therefore a good candidate for the demonstration of the NX .Nastran automatic restart capability (see Restarts ). A comparison of the HA146B and HA146D input files shows that the structure, aerodynamic models, and aerodynamic hard points are identical. The same boundary conditions are applied so that the eigenanalysis results (compare Listing 9-6 and Listing 9-12) are identical. Only the loadings differ with HA146B performing a transient roll analysis while the present analysis is for frequency response. The HA146D example can therefore be run as a “restart” of the a HA146B run that has saved its database. The job submittal for the cold start is: nast68 ha146b scr=no

Listing 9-13 shows the input data file for the restart run and it is seen that the Executive and Case Control Sections are identical to the HA146D file shown in Listing 9-10 with a RESTART command statement added at the top of the listing to indicate that the run is a restart and that it is to use the first “version” of data contained on the database. The Bulk Data Section is minimal in this run and first deletes the loading of the HA146B run by referencing the sorted Bulk Data echo line numbers of Listing 9-5 (the deletion of entries 97 through 101 removes TABLED1, TLOAD1, and TSTEP entries while the deletion of entry 45 removes the FREQ1 request). The subsequent RLOAD1, TABLED1 and FREQ1 entries define the new frequency response loading and are those used for HA146D. The restart run is submitted using nast68 ha146d_rst scr=no dbs=ha146b

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The sorted Bulk Data listing is shown in Listing 9-14. The X-Y output summary from the restart is shown in Listing 9-15, and these results are identical to those shown in Listing 9-12. N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN,HA146D $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ $ ANTISYMMETRIC RESPONSE TO A $ $ HARMONIC AILERON LOADING $ $ $ $ SOLUTION FREQUENCY RESPONSE ANALYSIS USING $ $ DOUBLET LATTICE METHOD AERODYNAMICS $ $ AT MACH NO. OF 0.0 $ $ $ $ OUTPUT XY PLOT OF WING ROOT BENDING MOMENT $ $ $ $$$$$$$$ $$$$$$$ TIME 10 $ CPU TIME IN MINUTES SOL 146 $ AEROELASTIC RESPONSE CEND BAH WING DYNAMIC FREQUENCY RESPONSE HA146D PAGE 3 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = BAH WING DYNAMIC FREQUENCY RESPONSE HA146D 2 SUBTI = ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO 3 LABEL = UNIT HARMONIC AILERON ROTATION 4 ECHO = BOTH 5 SPC = 13 $ BOUNDARY CONDITIONS (ANTISYMMETRIC) 6 MPC = 1 $ CONTROL SURFACE RELATIVE MOTION 7 METHOD = 10 $ MODIFIED-GIVENS EIGENVALUE METHOD 8 K2PP = ENFORCE $ EPOINT ADDED VIA DMIG 9 SDAMP = 2000 $ STRUCTURAL DAMPING (3 PERCENT) 10 DLOAD = 1000 $ FREQUENCY DEPENDENT LOAD 11 FREQ = 40 $ FREQUENCY LIST 12 OUTPUT 13 SET 1 = 11 14 SPCF(PHASE) = 1 $ SINGLE POINT CONSTRAINT FORCES 15 OUTPUT(XYOUT) $ XY PLOTTING PACKAGE 16 CSCALE 2.1 17 PLOTTER NASTRAN 18 TRIGHT TICS = -1 19 BRIGHT TICS = -1 20 XTGRID = YES 21 YTGRID = YES 22 XBGRID = YES 23 YBGRID = YES 24 YTTITLE = MAGNITUDE 25 YBTITLE = PHASE 26 XTITLE = FREQUENCY (HZ) 27 CURVELINESYMBOL = 6 28 TCURVE = WING ROOT BENDING MOMENT 29 YTITLE = SPC FORCE OF GRID 11 R3 30 XYPLOT SPCF / 11(R3RM,R3IP) 31 TCURVE = WING ROOT TORQUE 32 YTITLE = SPC FORCE OF GRID 11 R2 33 XYPLOT SPCF / 11(R2RM,R2IP) 34 BEGIN BULK BAH WING DYNAMIC FREQUENCY RESPONSE HA146D PAGE 4 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . INCLUDE BAH_STRUCT.DAT $ INCLUDE BAH_MASS.DAT $ INCLUDE BAH_AILERON $ $ TABDMP1 DEFINES DAMPING COEFFICIENTS (G1) VERSES FREQUENCY (F1). $ (3 PERCENT STRUCTURAL DAMPING IS SPECIFIED OVER THE ENTIRE FREQUENCY $ RANGE. VALUES ARE LINEARLY INTERPOLATED AND EXTRAPOLATED. A 3 $ PERCENT STRUCTURAL DAMPING IS ROUGHLY EQUIVALENT TO 1.5 PERCENT $ CRITICAL DAMPING). $ $ ID TYPE TABDMP1 2000 G +T2000 $ F1 G1 F2 G2 "ENDT" +T2000 0. .03 10. .03 ENDT $ $ * * CONSTRAINTS * * $ $ THE SPC ENTRY CONSTRAINS DOFS. $ $ SID G C D SPC 13 11 35 $

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$ * * * AERODYNAMIC DATA * * * $ $ (THIS MODEL USES THE LB-IN-SEC SYSTEM) $ $ * * ELEMENT GEOMETRY * * $ $ THE AERO ENTRY DEFINES BASIC AERODYNAMIC PARAMETERS. ACSID IS THE $ AERO COORDINATE SYSTEM. VELOCITY. REFC IS THE REFERENECE $ COORDINATE SYSTEM. RHOREF IS REFERENCE DENSITY. SYMXZ AND $ SYMXY ARE SYMMETRY KEYS. $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 1 8360. 131.25 1.1468-7-1 $ INCLUDE BAH_AERO58.DAT $ $ THE SPLINE1 ENTRY SPECIFIES A SURFACE SPLINE FOR INTERPOLATION $ OVER A REGION (BOX1 THRU BOX2) OF A CAERO PANEL AND THE $ STRUCTURAL GRIDS LISTED ON A SET1 FLAGGED BY SETG. DZ IS $ A LINEAR ATTACHMENT FLEXIBLITY. $ $ EID CAERO BOX1 BOX2 SETG DZ SPLINE1 104 2001 2005 2018 15 $ $ THE SET1 ENTRY DEFINES THE STRUCTURAL GRID POINTS USED FOR $ INTERPOLATION BY THE SPLINE. $ $ SID G1 G2 G3 G4 G5 G6 SET1 15 8 10 12 $ $ PARAM,M SPECIFIES MACH NUMBER. $ PARAM M 0.0 $ $ PARAM,Q SPECIFIES DYNAMIC PRESSURE. $ PARAM Q 4.00747 $ $ THE MKAERO1 ENTRY DEFINES COMBINATIONS OF MACH NUMBER AND REDUCED $ FREQUENCY EACH OF WHICH WILL BE USED TO GENERATE A MATRIX OF $ GENERALIZED AERODYNAMIC FORCES. $ $ M1 M2 M3 M4 ETC MKAERO1 0.0 +MK $ K1 K2 K3 K4 ETC +MK 0.001 0.02 0.10 0.50 $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ THE EPOINT ENTRY DEFINES AN "EXTRA" POINT IN THE STRUCTURAL MODEL. $ THIS ONE REPRESENTS A LAGRANGE MULTIPLIER FOR THE AILERON HINGE MOMENT. $ EPOINT 115 $ $ * * DYNAMIC LOAD AND RESPONSE DATA * * $ $ RLOAD1 DEFINES A FREQUENCY DEPENDENT DYNAMIC LOAD OR ENFORCED MOTION. $ LISTED ARE THE ID, DAREA ID, DELAY ID, DPHASE ID, AND TABLEDI IDS. $ $ SID DAREA DELAY DPHASE TC TD RLOAD1 1000 1001 1002 1003 $ $ DAREA DEFINES THE DOF WHERE THE LOAD IS APPLIED AND A SCALE FACTOR. $ $ SID P C A DAREA 1001 115 0 -1.0 $ $ TABLED1 DEFINES A TABULAR FUNCTION OF A TIME-DEPENDENT LOAD. $ $ SID TABLED1 1003 +T1003 $ X1 Y1 X2 Y2 ETC. +T1003 0. 1. 5. 1. 5. .0 10. .0 +T1003A +T1003A ENDT $ $ THE DELAY ENTRY SPECIFIES THE GRID POINT, DISPLACEMENT COMPONENT $ AND THE TIME DELAY OF ONSET OF THE FORCING FUNCTION. $ $ SID P C T DELAY 1002 115 0 .2 $ $ $ * * * USER SUPPLIED DIRECT MATRIX INPUT DATA * * * $ $ TIN=1 SPECIFIES THAT THE ELEMENTS WILL BE REAL AND SINGLE PRE$ CISION, TOUT=0 SPECIFIES THAT THE TYPE OF MATRIX TO BE GENERATED $ WILL BE DETERMINED INTERNALLY, POLAR=0 SPECIFIES THAT THE INPUT $ WILL BE IN ARGAND DIAGRAM FORM (X,IY). $ $ NAME "0" IFO TIN TOUT POLAR NCOL DMIG ENFORCE 0 1 1 0 $

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$ G2 AND C2 SPECIFY THAT A PITCHING MOMENT OF 1.0 IN LB BE APPLIED $ AT THE HINGE-LINE OF THE AILERON. SEE P. 3.5-9 OF THE "HANDBOOK $ FOR DYNAMIC ANALYSIS" FOR A DISCUSSION OF LAGRANGE MULTIPLIERS. $ THE MINUS SIGN ARISES FROM THE DEFINITION OF A POSITIVE AILERON $ DEFLECTION, I.E., TRAILING EDGE DOWN ON THE STARBOARD WING. $ $ NAME G1 C1 G2 C2 X21 DMIG ENFORCE 12 5 115 0 1. $ $ NAME G2 C2 X22 G1 C1 X12 DMIG ENFORCE 115 0 12 5 -1. $ $* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * $ $ EIGR DEFINES EIGENVALUE EXTRACTION DATA. LISTED ARE THE $ EIGENVALUE EXTRACTION METHOD, FREQUENCY RANGE, NUMBER OF $ EXPECTED AND DESIRED ROOTS AND THE METHOD OF NORMALIZATION. $ $ SID METHOD F1 F2 NE ND EIGR 10 MGIV 12 +EIGR $ NORM +EIGR MAX $ $ FREQ1 DEFINES THE SET OF FREQUENCIES USED TO OBTAIN $ THE FREQUENCY RESPONSE SOLUTION. LISTED ARE THE STARTING $ FREQUENCY, FREQUENCY INCREMENT AND NUMBER OF INCREMENTS. $ $ SID F1 DF NDF FREQ1 40 0. .1 50 $ ENDDATA INPUT BULK DATA CARD COUNT = 432

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Listing 9-10. Input Data for Frequency Response to Oscillating Aileron

9-48

BAH WING DYNAMIC FREQUENCY RESPONSE HA146D PAGE 13 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 1AEFACT 1 0. .09 .21 .33 .45 .56 .66 +AE1 2+AE1 .74 3AEFACT 2 .74 .82 .90 .974 4AEFACT 3 .974 1.00 5AEFACT 4 0. .1875 .375 .625 .750 .875 1.00 6AERO 1 8360. 131.25 1.1468-7-1 7CAERO1 1001 1000 0 5 1 1 +CA1 8+CA1 78.75 0. 0. 225. 35. 500. 0. 100. 9CAERO1 2001 1000 0 2 4 1 +CA2 10+CA2 78.75 0. 0. 225. 35. 500. 0. 100. 11CAERO1 3001 1000 0 5 3 1 +CA3 12+CA3 78.75 0. 0. 225. 35. 500. 0. 100. 13CELAS2 3 5142661.12 5 14CMASS2 121 5248.7 1 3 15CMASS2 122 134.9 1 3 2 3 16CMASS2 123 790.3 2 3 17CMASS2 341 9727. 3 3 18CMASS2 342 11005. 3 3 4 3 19CMASS2 343 473. 4 3 20CMASS2 561 3253.6 5 3 21CMASS2 562 -139.7 5 3 6 3 22CMASS2 563 946.3 6 3 23CMASS2 781 2617.8 7 3 24CMASS2 782 21. 7 3 8 3 25CMASS2 783 782.3 8 3 26CMASS2 9101 494.8 9 3 27CMASS2 9102 -7.3 9 3 10 3 28CMASS2 9103 185.2 10 3 29CONM1 1 11 +51 30+51 17400. 4.37+7 +52 31+52 4.35+09 32CONM1 2 12 +AIL1 33+AIL1 0.0 0.0 +AIL2 34+AIL2 13970.5 35CORD2R 1 0. 0. 0. 0. 0. -1. +C1 36+C1 -1. 0. 0. 37DAREA 1001 115 0 -1.0 38DELAY 1002 115 0 .2 39DMIG ENFORCE 0 1 1 0 40DMIG ENFORCE 12 5 115 0 1. 41DMIG ENFORCE 115 0 12 5 -1. 42EIGR 10 MGIV 12 +EIGR 43+EIGR MAX 44EPOINT 115 45FREQ1 40 0. .1 50 46GENEL 432 1 3 2 3 3 3 +01 47+01 4 3 5 3 6 3 7 3 +02 48+02 8 3 9 3 10 3 +03 49+03 UD 11 3 11 4 11 5 +04 50+04 11 6 +05 51+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 52+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 53+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 54+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 55+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 56+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 57+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 58+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 59+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 60+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 61+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 62+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 63+17 413.0 64GRID 1 20.25 90. 12456 65GRID 2 -81. 90. 12456 66GRID 3 17.85 186. 12456 67GRID 4 -71.4 186. 12456 68GRID 5 15.8 268. 12456 69GRID 6 -63.2 268. 12456 70GRID 7 13.3 368. 12456 71GRID 8 -53.2 368. 12456 72GRID 9 11.05 458. 12456 73GRID 10 -44.2 458. 12456 74GRID 11 0.0 0. 126 75GRID 12 -86.45 368. 1246 76MKAERO1 0.0 +MK 77+MK 0.001 0.02 0.10 0.50 78MPC 1 12 3 -1.0 8 3 1.5 +MPC1 79+MPC1 7 3 -0.5 12 5 33.25 80PAERO1 1000 81PARAM GRDPNT 11 82PARAM M 0.0 83PARAM Q 4.00747 84PARAM WTMASS .0025907 85RLOAD1 1000 1001 1002 1003 NX Nastran Aeroelastic Analysis User’s14 Guide 1 86SET1 THRU 11 87SET1 15 8 10 12 88SPC 13 11 35

.

Dynamic Aeroelastic Response Analysis

919293949596979899100-

+SP1 SPLINE2 +SP2 SPLINE2 +SP3 TABDMP1 +T2000 TABLED1 +T1003 +T1003A ENDDATA TOTAL COUNT=

-1.0 102 -1.0 103 -1.0 2000 0. 1003 0. ENDT

-1.0 2001 -1.0 3001 -1.0 G .03 1.

2001

2016

14

0.

1.

0

+SP2

3001

3005

14

0.

1.

0

+SP3

10.

.03

ENDT

5.

1.

5.

+T2000 .0

10.

.0

+T1003 +T1003A

101

Listing 9-11. Sorted Bulk Data for Frequency Response to Oscillating Aileron

MODE NO. 1 2 3 4 5 6 7 8 9 10 11 12

PAGE 19 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION R E A L E I G E N V A L U E S EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED ORDER MASS STIFFNESS 1 -6.892265E-13 8.301967E-07 1.321299E-07 1.667624E+01 -1.149371E-11 2 4.945913E+02 2.223941E+01 3.539512E+00 4.167461E+01 2.061190E+04 3 8.508833E+02 2.916990E+01 4.642534E+00 7.082290E+00 6.026202E+03 4 4.434137E+03 6.658932E+01 1.059802E+01 5.831286E+00 2.585672E+04 5 8.732797E+03 9.344943E+01 1.487294E+01 4.022015E+00 3.512344E+04 7 1.501493E+04 1.225354E+02 1.950212E+01 1.220782E+02 1.832996E+06 9 1.766970E+04 1.329274E+02 2.115605E+01 3.908876E+00 6.906864E+04 8 2.409598E+04 1.552288E+02 2.470543E+01 3.522202E+00 8.487088E+04 11 4.211864E+04 2.052283E+02 3.266309E+01 3.142603E+00 1.323622E+05 10 6.020845E+04 2.453741E+02 3.905250E+01 1.016245E+00 6.118652E+04 6 9.304316E+04 3.050298E+02 4.854699E+01 8.852293E+00 8.236453E+05 12 1.420885E+05 3.769462E+02 5.999286E+01 3.619337E+01 5.142661E+06 POINT-ID = 11 C O M P L E X F O R C E S O F S I N G L E P O I N T C O N S T R A I N T (MAGNITUDE/PHASE)

FREQUENCY 0.0

TYPE G

1.000000E-01

G

2.000000E-01

G

3.000000E-01

G

4.000000E-01

G

5.000000E-01

G

6.000000E-01

G

7.000000E-01

G

8.000000E-01

G

9.000000E-01

G

1.000000E+00

G

1.100000E+00

G

1.200000E+00

G

1.300000E+00

G

1.400000E+00

G

1.500000E+00

G

1.600000E+00

G

1.700000E+00

G

1.800000E+00

G

1.900000E+00

G

2.000000E+00

G

2.100000E+00

G

2.200000E+00

G

2.300000E+00

G

2.400000E+00

G

2.500000E+00

G

2.600000E+00

G

2.700000E+00

G

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T3 2.885785E+05 0.0 3.462686E+04 358.9044 3.715200E+04 355.5831 4.003538E+04 349.8983 4.263010E+04 342.7689 4.477215E+04 334.9677 4.651321E+04 326.9194 4.795612E+04 318.8221 4.919577E+04 310.7581 5.030629E+04 302.7567 5.134258E+04 294.8237 5.234507E+04 286.9551 5.334400E+04 279.1432 5.436272E+04 271.3799 5.542002E+04 263.6571 5.653175E+04 255.9673 5.771199E+04 248.3040 5.897386E+04 240.6612 6.033011E+04 233.0336 6.179361E+04 225.4163 6.337767E+04 217.8048 6.509627E+04 210.1947 6.696416E+04 202.5821 6.899677E+04 194.9630 7.120993E+04 187.3338 7.361891E+04 179.6911 7.623647E+04 172.0331 7.906870E+04

R1 R2 R3 0.0 2.497663E+06 4.209172E+06 0.0 180.0000 180.0000 0.0 5.341617E+06 1.656624E+06 0.0 169.0537 187.0946 0.0 5.185181E+06 1.976318E+06 0.0 158.8335 186.7975 0.0 4.994818E+06 2.307983E+06 0.0 149.6563 180.9410 0.0 4.818837E+06 2.585938E+06 0.0 141.4345 172.6773 0.0 4.680143E+06 2.803856E+06 0.0 133.9086 163.6324 0.0 4.583896E+06 2.973687E+06 0.0 126.8211 154.4629 0.0 4.527535E+06 3.109103E+06 0.0 119.9813 145.4042 0.0 4.506436E+06 3.221133E+06 0.0 113.2641 136.5240 0.0 4.516295E+06 3.317836E+06 0.0 106.5927 127.8266 0.0 4.553872E+06 3.404923E+06 0.0 99.9220 119.2944 0.0 4.617120E+06 3.486451E+06 0.0 93.2259 110.9041 0.0 4.705084E+06 3.565345E+06 0.0 86.4898 102.6332 0.0 4.817798E+06 3.643771E+06 0.0 79.7055 94.4615 0.0 4.956208E+06 3.723377E+06 0.0 72.8680 86.3716 0.0 5.122163E+06 3.805466E+06 0.0 65.9741 78.3490 0.0 5.318473E+06 3.891096E+06 0.0 59.0207 70.3810 0.0 5.549037E+06 3.981153E+06 0.0 52.0046 62.4572 0.0 5.819067E+06 4.076394E+06 0.0 44.9216 54.5683 0.0 6.135434E+06 4.177455E+06 0.0 37.7659 46.7062 0.0 6.507188E+06 4.284857E+06 0.0 30.5295 38.8639 0.0 6.946306E+06 4.398957E+06 0.0 23.2014 31.0349 0.0 7.468850E+06 4.519882E+06 0.0 15.7665 23.2140 0.0 8.096700E+06 4.647391E+06 0.0 8.2037 15.3965 0.0 8.860268E+06 4.780634E+06 0.0 0.4834 7.5795 0.0 9.802894E+06 4.917747E+06 0.0 352.5630 359.7631 0.0 1.098824E+07 5.055111E+06 0.0 344.3796 351.9530 0.0 1.251318E+07 5.186022E+06 9-49 NX Nastran Aeroelastic Analysis User’s Guide

Dynamic Aeroelastic Response Analysis

Chapter 9

0.0 164.3605 0.0 335.8372 344.1685 0.0 8.210698E+04 0.0 1.453150E+07 5.298224E+06 0.0 156.6807 0.0 326.7820 336.4601 2.900000E+00 G 0.0 8.531131E+04 0.0 1.729956E+07 5.369289E+06 0.0 149.0185 0.0 316.9529 328.9594 3.000000E+00 G 0.0 8.857972E+04 0.0 2.126843E+07 5.358764E+06 0.0 141.4457 0.0 305.8746 322.0317 3.100000E+00 G 0.0 9.170979E+04 0.0 2.726900E+07 5.202995E+06 0.0 134.1740 0.0 292.6027 316.8169 3.200000E+00 G 0.0 9.454148E+04 0.0 3.678848E+07 4.901936E+06 0.0 127.8587 0.0 275.0898 317.2259 3.300000E+00 G 0.0 9.895744E+04 0.0 5.121748E+07 5.386076E+06 0.0 124.1503 0.0 248.9234 330.4797 3.400000E+00 G 0.0 1.153433E+05 0.0 6.292264E+07 9.125889E+06 0.0 121.4842 0.0 209.5719 331.7407 3.500000E+00 G 0.0 1.397750E+05 0.0 5.541844E+07 1.288197E+07 0.0 111.2863 0.0 169.1243 310.0754 3.600000E+00 G 0.0 1.569750E+05 0.0 4.209812E+07 1.450235E+07 0.0 99.1698 0.0 141.4372 290.7008 3.700000E+00 G 0.0 1.727535E+05 0.0 3.290603E+07 1.566987E+07 0.0 88.2349 0.0 122.6299 276.0620 3.800000E+00 G 0.0 1.914614E+05 0.0 2.721831E+07 1.709938E+07 0.0 77.8461 0.0 107.9484 263.6051 3.900000E+00 G 0.0 2.153090E+05 0.0 2.368604E+07 1.902751E+07 0.0 67.3816 0.0 95.0518 251.8620 4.000000E+00 G 0.0 2.465869E+05 0.0 2.153033E+07 2.165180E+07 0.0 56.3640 0.0 82.6988 239.9876 4.100000E+00 G 0.0 2.885493E+05 0.0 2.035072E+07 2.525212E+07 0.0 44.2614 0.0 69.9875 227.2745 4.200000E+00 G 0.0 3.458575E+05 0.0 1.993973E+07 3.023772E+07 0.0 30.2835 0.0 55.9403 212.8407 4.300000E+00 G 0.0 4.235863E+05 0.0 2.011624E+07 3.706418E+07 0.0 13.1206 0.0 39.1570 195.3249 4.400000E+00 G 0.0 5.187861E+05 0.0 2.037909E+07 4.549716E+07 0.0 350.7889 0.0 17.6234 172.7120 4.500000E+00 G 0.0 5.933007E+05 0.0 1.930026E+07 5.221080E+07 0.0 321.9275 0.0 349.9944 143.6212 4.600000E+00 G 0.0 5.778406E+05 0.0 1.554109E+07 5.107458E+07 0.0 290.1584 0.0 319.9532 111.6617 4.700000E+00 G 0.0 4.870178E+05 0.0 1.078277E+07 4.327256E+07 0.0 262.7525 0.0 294.8864 84.0952 4.800000E+00 G 0.0 3.911368E+05 0.0 7.079736E+06 3.496078E+07 0.0 241.8658 0.0 277.1436 63.0714 4.900000E+00 G 0.0 3.165107E+05 0.0 4.640559E+06 2.847766E+07 0.0 225.7154 0.0 265.2546 46.8027 5.000000E+00 G 0.0 1.309741E+05 0.0 1.539469E+06 1.186908E+07 0.0 212.4672 0.0 257.8848 33.4515 X Y - O U T P U T S U M M A R Y ( R E S P O N S E ) SUBCASE CURVE FRAME XMIN-FRAME/ XMAX-FRAME/ YMIN-FRAME/ X FOR YMAX-FRAME/ X FOR ID TYPE NO. CURVE ID. ALL DATA ALL DATA ALL DATA YMIN ALL DATA YMAX 1 SPCF 1 11( 8,--) 0.000000E+00 5.000000E+00 1.656624E+06 1.000000E-01 5.221080E+07 4.500000E+00 0.000000E+00 5.000000E+00 1.656624E+06 1.000000E-01 5.221080E+07 4.500000E+00 1 SPCF 1 11(--, 14) 0.000000E+00 5.000000E+00 7.579535E+00 2.400000E+00 3.597631E+02 2.500000E+00 0.000000E+00 5.000000E+00 7.579535E+00 2.400000E+00 3.597631E+02 2.500000E+00 1 SPCF 2 11( 7,--) 0.000000E+00 5.000000E+00 1.539469E+06 5.000000E+00 6.292264E+07 3.400000E+00 0.000000E+00 5.000000E+00 1.539469E+06 5.000000E+00 6.292264E+07 3.400000E+00 1 SPCF 2 11(--, 13) 0.000000E+00 5.000000E+00 4.834088E-01 2.400000E+00 3.525630E+02 2.500000E+00 0.000000E+00 5.000000E+00 4.834088E-01 2.400000E+00 3.525630E+02 2.500000E+00 2.800000E+00

G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Listing 9-12. Output for Frequency Response to Oscillating Aileron N A S T R A N

F I L E

M A N A G E M E N T

S E C T I O N

E C H O

RESTART VERSION=1,KEEP PAGE N A S T R A N

9-50

E X E C U T I V E

2 C O N T R O L

D E C K

E C H O

ID NXN,HA146DR $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE $$$$$$$$ $ $ $ MODEL DESCRIPTION BAH JET TRANSPORT WING EXAMPLE $ $ ANTISYMMETRIC RESPONSE TO A $ $ HARMONIC AILERON LOADING $ $ $ $ SOLUTION FREQUENCY RESPONSE ANALYSIS USING $ $ DOUBLET LATTICE METHOD AERODYNAMICS $ $ AT MACH NO. OF 0.0 $ $ $ $ OUTPUT XY PLOT OF WING ROOT BENDING MOMENT $ $ $ $$$$$$$$ $$$$$$$ TIME 10 $ CPU TIME IN MINUTES SOL 146 $ AEROELASTIC RESPONSE CEND BAH WING DYNAMIC FREQUENCY RESPONSE HA146DR PAGE 3 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = BAH WING DYNAMIC FREQUENCY RESPONSE HA146DR NX Nastran Aeroelastic Analysis User’s Guide 2 SUBTI = ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO

Dynamic Aeroelastic Response Analysis

3 LABEL = UNIT HARMONIC AILERON ROTATION 4 ECHO = BOTH 5 SPC = 13 $ BOUNDARY CONDITIONS (ANTISYMMETRIC) 6 MPC = 1 $ CONTROL SURFACE RELATIVE MOTION 7 METHOD = 10 $ MODIFIED-GIVENS EIGENVALUE METHOD 8 K2PP = ENFORCE $ EPOINT ADDED VIA DMIG 9 SDAMP = 2000 $ STRUCTURAL DAMPING (3 PERCENT) 10 DLOAD = 1000 $ FREQUENCY DEPENDENT LOAD 11 FREQ = 40 $ FREQUENCY LIST 12 OUTPUT 13 SET 1 = 11 14 SPCF(PHASE) = 1 $ SINGLE POINT CONSTRAINT FORCES 15 OUTPUT(XYOUT) $ XY PLOTTING PACKAGE 16 CSCALE 2.1 17 PLOTTER NASTRAN 18 TRIGHT TICS = -1 19 BRIGHT TICS = -1 20 XTGRID = YES 21 YTGRID = YES 22 XBGRID = YES 23 YBGRID = YES 24 YTTITLE = MAGNITUDE 25 YBTITLE = PHASE 26 XTITLE = FREQUENCY (HZ) 27 CURVELINESYMBOL = 6 28 TCURVE = WING ROOT BENDING MOMENT 29 YTITLE = SPC FORCE OF GRID 11 R3 30 XYPLOT SPCF / 11(R3RM,R3IP) 31 TCURVE = WING ROOT TORQUE 32 YTITLE = SPC FORCE OF GRID 11 R2 33 XYPLOT SPCF / 11(R2RM,R2IP) 34 BEGIN BULK BAH WING DYNAMIC FREQUENCY RESPONSE HA146DR PAGE 4 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION I N P U T B U L K D A T A D E C K E C H O . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . / 97 101 / 45 $ $ * * DYNAMIC LOAD AND RESPONSE DATA * * $ $ RLOAD1 DEFINES A FREQUENCY DEPENDENT DYNAMIC LOAD OR ENFORCED MOTION. $ LISTED ARE THE ID, DAREA ID, DELAY ID, DPHASE ID, AND TABLEDI IDS. $ $ SID DAREA DELAY DPHASE TC TD RLOAD1 1000 1001 1002 1003 $ $ $ TABLED1 DEFINES A TABULAR FUNCTION OF A TIME-DEPENDENT LOAD. $ $ SID TABLED1 1003 +T1003 $ X1 Y1 X2 Y2 ETC. +T1003 0. 1. 5. 1. 5. .0 10. .0 +T1003A +T1003A ENDT $ $ FREQ1 DEFINES THE SET OF FREQUENCIES USED TO OBTAIN $ THE FREQUENCY RESPONSE SOLUTION. LISTED ARE THE STARTING $ FREQUENCY, FREQUENCY INCREMENT AND NUMBER OF INCREMENTS. $ $ SID F1 DF NDF FREQ1 40 0. .1 50 $ $ ENDDATA INPUT BULK DATA CARD COUNT = 29

Listing 9-13. Input Data for Frequency Response to Oscillating Aileron via Restart BAH WING DYNAMIC FREQUENCY RESPONSE HA146DR PAGE 5 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 1AEFACT 1 0. .09 .21 .33 .45 .56 .66 2+AE1 .74 3AEFACT 2 .74 .82 .90 .974 4AEFACT 3 .974 1.00 5AEFACT 4 0. .1875 .375 .625 .750 .875 1.00 6AERO 1 8360. 131.232 1.1468-7-1 7CAERO1 1001 1000 0 5 1 1 8+CA1 78.75 0. 0. 225. 35. 500. 0. 100. 9CAERO1 2001 1000 0 2 4 1 10+CA2 78.75 0. 0. 225. 35. 500. 0. 100. 11CAERO1 3001 1000 0 5 3 1 12+CA3 78.75 0. 0. 225. 35. 500. 0. 100. 13CELAS2 3 5142661.12 5 14CMASS2 121 5248.7 1 3 15CMASS2 122 134.9 1 3 2 3 16CMASS2 123 790.3 2 3 17CMASS2 341 9727. 3 3 NX Nastran Aeroelastic Analysis User’s Guide 18CMASS2 342 11005. 3 3 4 3 19CMASS2 343 473. 4 3

.. 10 +AE1

+CA1 +CA2 +CA3

9-51

.

Chapter 9

Dynamic Aeroelastic Response Analysis

20CMASS2 561 3253.6 5 3 21CMASS2 562 -139.7 5 3 6 3 22CMASS2 563 946.3 6 3 23CMASS2 781 2617.8 7 3 24CMASS2 782 21. 7 3 8 3 25CMASS2 783 782.3 8 3 26CMASS2 9101 494.8 9 3 27CMASS2 9102 -7.3 9 3 10 3 28CMASS2 9103 185.2 10 3 29CONM1 1 11 +51 30+51 17400. 4.37+7 +52 31+52 4.35+09 32CONM1 2 12 +AIL1 33+AIL1 0.0 0.0 +AIL2 34+AIL2 13970.5 35CORD2R 1 0. 0. 0. 0. 0. -1. +C1 36+C1 -1. 0. 0. 37DAREA 1001 115 0 -1.0 38DELAY 1002 115 0 .2 39DMIG ENFORCE 0 1 1 0 40DMIG ENFORCE 12 5 115 0 1. 41DMIG ENFORCE 115 0 12 5 -1. 42EIGR 10 MGIV 12 +EIGR 43+EIGR MAX 44EPOINT 115 45FREQ1 40 0. .1 50 46GENEL 432 1 3 2 3 3 3 +01 47+01 4 3 5 3 6 3 7 3 +02 48+02 8 3 9 3 10 3 +03 49+03 UD 11 3 11 4 11 5 +04 50+04 11 6 +05 BAH WING DYNAMIC FREQUENCY RESPONSE HA146DR PAGE 6 ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 51+05 Z 8.7172-61.3361-61.2778-56.2720-61.6251-51.0492-52.0478-5+06 52+06 1.5630-52.4285-52.0403-53.0861-56.2720-63.2297-51.0492-53.3529-5+07 53+07 1.5630-53.5021-52.0257-53.5785-52.7732-51.5726-54.8255-53.7628-5+08 54+08 7.3284-56.4338-59.5810-58.8378-56.3749-53.7628-58.0136-56.4338-5+09 55+09 1.0012-48.8378-51.1811-41.2758-41.1344-41.9350-41.8160-42.5283-4+10 56+10 2.4294-41.6999-41.8160-42.2920-42.4294-42.8249-43.6862-43.5052-4+11 57+11 5.2675-45.1171-44.2292-45.1171-45.7187-48.4840-48.2340-49.2340-4+12 58+12 S 1.0 90.0 -20.25 45.0 1.0 90.0 81.0 +13 59+13 45.0 1.0 186.0 -17.85 141.0 1.0 186.0 71.4 +14 60+14 141.0 1.0 268.0 -15.80 223.0 1.0 268.0 63.2 +15 61+15 223.0 1.0 368.0 -13.30 323.0 1.0 368.0 53.2 +16 62+16 323.0 1.0 458.0 -11.05 413.0 1.0 458.0 44.2 +17 63+17 413.0 64GRID 1 20.25 90. 12456 65GRID 2 -81. 90. 12456 66GRID 3 17.85 186. 12456 67GRID 4 -71.4 186. 12456 68GRID 5 15.8 268. 12456 69GRID 6 -63.2 268. 12456 70GRID 7 13.3 368. 12456 71GRID 8 -53.2 368. 12456 72GRID 9 11.05 458. 12456 73GRID 10 -44.2 458. 12456 74GRID 11 0.0 0. 126 75GRID 12 -86.45 368. 1246 76MKAERO1 0.0 +MK 77+MK 0.001 0.02 0.10 0.50 78MPC 1 12 3 -1.0 8 3 1.5 +MPC1 79+MPC1 7 3 -0.5 12 5 33.25 80PAERO1 1000 81PARAM GRDPNT 11 82PARAM LMODES 12 83PARAM Q 4.00747 84PARAM WTMASS .0025907 85RLOAD1 1000 1001 1002 1003 86SET1 14 1 THRU 11 87SET1 15 8 10 12 88SPC 13 11 35 89SPLINE1 104 2001 2005 2018 15 90SPLINE2 101 1001 1001 1035 14 0. 1. 0 +SP1 91+SP1 -1.0 -1.0 92SPLINE2 102 2001 2001 2016 14 0. 1. 0 +SP2 93+SP2 -1.0 -1.0 94SPLINE2 103 3001 3001 3005 14 0. 1. 0 +SP3 95+SP3 -1.0 -1.0 96TABDMP1 2000 +T2000 97+T2000 0. .03 10. .03 ENDT 98TABLED1 1003 +T1003 99+T1003 0. 1. 5. 1. 5. .0 10. .0 +T1003A 100+T1003A ENDT ENDDATA TOTAL COUNT= 101

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Listing 9-14. Sorted Bulk Data for Frequency Response to Oscillating Aileron via Restart BAH WING DYNAMIC FREQUENCY RESPONSE HA146DR ANTISYMMETRIC RESPONSE, DOUBLET LATTICE AERO UNIT HARMONIC AILERON ROTATION POINT-ID = 11 C O M P L E X F O R C E S O F S I N G L E P O I N T (MAGNITUDE/PHASE) FREQUENCY 0.0

TYPE G

1.000000E-01 G 2.000000E-01 G 3.000000E-01 G 4.000000E-01 G 5.000000E-01 G 6.000000E-01 G 7.000000E-01 G 8.000000E-01 G 9.000000E-01 G 1.000000E+00 G 1.100000E+00 G 1.200000E+00 G 1.300000E+00 G 1.400000E+00 G 1.500000E+00 G 1.600000E+00 G 1.700000E+00 G 1.800000E+00 G 1.900000E+00 G 2.000000E+00 G 2.100000E+00 G 2.200000E+00 G 2.300000E+00 G 2.400000E+00 G 2.500000E+00 G 2.600000E+00 G 2.700000E+00 G 2.800000E+00 G 2.900000E+00 G 3.000000E+00 G 3.100000E+00 G 3.200000E+00 G 3.300000E+00 G 3.400000E+00 G 3.500000E+00 G 3.600000E+00 G 3.700000E+00 G 3.800000E+00 G 3.900000E+00 G

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T3 2.885140E+05 0.0 3.462687E+04 358.9045 3.715199E+04 355.5831 4.003538E+04 349.8983 4.263010E+04 342.7689 4.477215E+04 334.9677 4.651322E+04 326.9194 4.795611E+04 318.8221 4.919577E+04 310.7581 5.030629E+04 302.7567 5.134258E+04 294.8237 5.234507E+04 286.9551 5.334401E+04 279.1433 5.436272E+04 271.3799 5.542002E+04 263.6571 5.653175E+04 255.9673 5.771199E+04 248.3040 5.897385E+04 240.6612 6.033011E+04 233.0336 6.179360E+04 225.4163 6.337767E+04 217.8048 6.509627E+04 210.1948 6.696414E+04 202.5822 6.899676E+04 194.9631 7.120991E+04 187.3338 7.361890E+04 179.6912 7.623645E+04 172.0331 7.906869E+04 164.3605 8.210696E+04 156.6807 8.531130E+04 149.0186 8.857969E+04 141.4457 9.170978E+04 134.1740 9.454145E+04 127.8586 9.895737E+04 124.1502 1.153431E+05 121.4843 1.397748E+05 111.2863 1.569748E+05 99.1699 1.727534E+05 88.2349 1.914612E+05 77.8461 2.153087E+05

R1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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C O N S T R A I N T R2 2.497399E+06 180.0000 5.341615E+06 169.0536 5.185183E+06 158.8336 4.994818E+06 149.6563 4.818835E+06 141.4345 4.680143E+06 133.9086 4.583895E+06 126.8211 4.527534E+06 119.9813 4.506437E+06 113.2641 4.516294E+06 106.5927 4.553871E+06 99.9220 4.617120E+06 93.2259 4.705084E+06 86.4898 4.817797E+06 79.7055 4.956208E+06 72.8680 5.122164E+06 65.9741 5.318474E+06 59.0207 5.549038E+06 52.0046 5.819067E+06 44.9216 6.135434E+06 37.7659 6.507188E+06 30.5295 6.946305E+06 23.2014 7.468850E+06 15.7665 8.096700E+06 8.2037 8.860268E+06 0.4834 9.802895E+06 352.5630 1.098824E+07 344.3796 1.251318E+07 335.8372 1.453150E+07 326.7820 1.729956E+07 316.9529 2.126843E+07 305.8746 2.726901E+07 292.6028 3.678849E+07 275.0900 5.121757E+07 248.9236 6.292292E+07 209.5720 5.541867E+07 169.1242 4.209824E+07 141.4371 3.290609E+07 122.6298 2.721834E+07 107.9483 2.368606E+07

R3 4.208535E+06 180.0000 1.656625E+06 187.0946 1.976318E+06 186.7975 2.307983E+06 180.9410 2.585938E+06 172.6773 2.803856E+06 163.6324 2.973687E+06 154.4629 3.109102E+06 145.4042 3.221134E+06 136.5241 3.317836E+06 127.8266 3.404923E+06 119.2944 3.486451E+06 110.9041 3.565346E+06 102.6332 3.643771E+06 94.4615 3.723378E+06 86.3716 3.805467E+06 78.3490 3.891096E+06 70.3811 3.981154E+06 62.4572 4.076394E+06 54.5683 4.177455E+06 46.7062 4.284857E+06 38.8639 4.398956E+06 31.0350 4.519881E+06 23.2140 4.647391E+06 15.3965 4.780633E+06 7.5796 4.917747E+06 359.7631 5.055110E+06 351.9530 5.186022E+06 344.1685 5.298223E+06 336.4601 5.369287E+06 328.9594 5.358762E+06 322.0317 5.202992E+06 316.8169 4.901927E+06 317.2258 5.386052E+06 330.4798 9.125871E+06 331.7410 1.288197E+07 310.0756 1.450235E+07 290.7009 1.566986E+07 276.0621 1.709937E+07 263.6051 1.902749E+07

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0.0 67.3816 0.0 95.0517 251.8620 0.0 2.465867E+05 0.0 2.153034E+07 2.165179E+07 0.0 56.3641 0.0 82.6988 239.9877 4.100000E+00 G 0.0 2.885489E+05 0.0 2.035072E+07 2.525210E+07 0.0 44.2615 0.0 69.9875 227.2746 4.200000E+00 G 0.0 3.458573E+05 0.0 1.993974E+07 3.023770E+07 0.0 30.2835 0.0 55.9403 212.8408 4.300000E+00 G 0.0 4.235860E+05 0.0 2.011624E+07 3.706416E+07 0.0 13.1207 0.0 39.1570 195.3250 4.400000E+00 G 0.0 5.187858E+05 0.0 2.037910E+07 4.549714E+07 0.0 350.7890 0.0 17.6234 172.7121 4.500000E+00 G 0.0 5.933006E+05 0.0 1.930028E+07 5.221080E+07 0.0 321.9276 0.0 349.9944 143.6213 4.600000E+00 G 0.0 5.778407E+05 0.0 1.554111E+07 5.107459E+07 0.0 290.1585 0.0 319.9532 111.6618 4.700000E+00 G 0.0 4.870179E+05 0.0 1.078278E+07 4.327257E+07 0.0 262.7526 0.0 294.8865 84.0953 4.800000E+00 G 0.0 3.911369E+05 0.0 7.079745E+06 3.496079E+07 0.0 241.8659 0.0 277.1437 63.0714 4.900000E+00 G 0.0 3.165107E+05 0.0 4.640564E+06 2.847766E+07 0.0 225.7155 0.0 265.2547 46.8027 5.000000E+00 G 0.0 1.309742E+05 0.0 1.539471E+06 1.186908E+07 0.0 212.4672 0.0 257.8848 33.4515 X Y - O U T P U T S U M M A R Y ( R E S P O N S E ) SUBCASE CURVE FRAME XMIN-FRAME/ XMAX-FRAME/ YMIN-FRAME/ X FOR YMAX-FRAME/ X FOR ID TYPE NO. CURVE ID. ALL DATA ALL DATA ALL DATA YMIN ALL DATA YMAX 1 SPCF 1 11( 8,--) 0.000000E+00 5.000000E+00 1.656625E+06 1.000000E-01 5.221080E+07 4.500000E+00 0.000000E+00 5.000000E+00 1.656625E+06 1.000000E-01 5.221080E+07 4.500000E+00 1 SPCF 1 11(--, 14) 0.000000E+00 5.000000E+00 7.579554E+00 2.400000E+00 3.597631E+02 2.500000E+00 0.000000E+00 5.000000E+00 7.579554E+00 2.400000E+00 3.597631E+02 2.500000E+00 1 SPCF 2 11( 7,--) 0.000000E+00 5.000000E+00 1.539471E+06 5.000000E+00 6.292292E+07 3.400000E+00 0.000000E+00 5.000000E+00 1.539471E+06 5.000000E+00 6.292292E+07 3.400000E+00 1 SPCF 2 11(--, 13) 0.000000E+00 5.000000E+00 4.834216E-01 2.400000E+00 3.525630E+02 2.500000E+00 0.000000E+00 5.000000E+00 4.834216E-01 2.400000E+00 3.525630E+02 2.500000E+00 4.000000E+00 G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Listing 9-15. Output for Frequency response to Oscillating Aileron via Restart

9.6 Subsonic Transient Response Analysis of a Sweptback Wing to an Impulsive Force Applied at the Tip (Example HA146E) This example is the same model of the sweptback wing that was used in the flutter analysis of Example HA145E with an impulsive load now applied at the wing tip. The flutter speed of this model, determined by Example HA145E, is 491 ft/s = 5892 in/s and is one of these velocities at which this analysis is performed. Analyses are also performed at velocities above and below the flutter speed. The structural model and the aerodynamic model are the same as those described in the flutter analysis of Example HA145E as are the root mounting conditions. A four mode solution is again used in order to have the identical flutter speed. The new loading entries are discussed below. The TLOAD1 10 entry specifies the DAREA 12, DELAY 13, and TABLED1 14 entries that define the impulsive loading. The loading is applied at GRID 40 at the tip trailing edge of the wing in the positive z-direction. A frequency interval of Δf = 5.0 Hz is selected and results in a period T = 1 / Δf = 0.2 seconds. A 0.1 second delay (i.e., half the period) is specified on the DELAY entry before the impulse is applied so that the response for a complete period need not be calculated; however, two full cycles are calculated here to illustrate the complete behavior of the Fourier solution in the three cases of stability (below the flutter speed), neutral stability (at the flutter speed), and instability (above the flutter speed). The unit impulse (the area under the curve is 1.0) is applied for 10−6 seconds with a force of 106 lbs. The unit impulse is followed by a second impulse that is equal in magnitude but opposite in sign. As explained in Example HA146A , certain output, such as stresses, will now tend to a zero value; however, in this example of a cantilever wing, the deflections will also approach zero. Three analyses are performed on the swept wing at three different speeds. The data input files are different only in the velocity on the AERO entry and in the corresponding dynamic pressure defined by PARAM,Q. The velocities are 450.0, 491.0 (the flutter speed), and 530.0 ft/s and the corresponding dynamic pressures based on the wind tunnel test density of 0.0023 slugs/cu. ft. are 1.617, 1.925, and 2.343 psi, respectively. The ENDDATA entry completes the Bulk Data Section.

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Case Control Commands The Case Control Section contains the TITLE, SUBTITLE, and LABEL commands. ECHO = BOTH echoes back all of the Bulk Data. SPC = 1 selects the boundary conditions, METHOD = 10 selects the EIGR entry for the vibration analysis, SDAMP = 2000 selects the damping, DLOAD = 10 selects the impulsive load, FREQ = 101 selects the frequency list and TSTEP = 201 selects the time history output. The DISP command specifies that displacements of GRID 40 be calculated and printed. Following OUTPUT(XYOUT) is an xy-plot request. Preceding the plot request are several optional statements that define the plot frame and plot titles. An XYPLOT is requested for the tip displacement of the wing at GRID 40 in the z-direction. The BEGIN BULK command completes the Case Control Section. The Executive Control Section lists the identification as ID NXN, HA146E, which denotes Problem No. HA146E in the Test Problem Library. TIME 10 limits the CPU to 10.0 minutes. SOL 146 is the Dynamic Aeroelastic Response solution sequence. The CEND statement completes the Executive Control Section. The input data are shown in Listing 9-16, the sorted Bulk Data are in Listing 9-17, and selected output are in Listing 9-18. The output is discussed briefly below. Figure 9-12 is from the analysis performed at 450 ft/s = 5400 in/s. Since there is a 0.1 sec delay, the first full period plotted is from 0.1 to 0.3 seconds. The responses from 0.0 to 0.1 seconds and from 0.2 to 0.3 seconds represent the same portions of separate periods. Figure 9-12 shows, that below the flutter speed, the tip displacement peaks immediately and then damps out to negligible values in about 0.1 sec. This type of stable behavior is expected below the flutter speed. Figure 9-13 is from the analysis performed at the flutter speed of 491 ft/s = 5892 in/s and shows that the same impulsive load first excites a stable mode, which damps out, followed by an unstable mode that appears as a divergent oscillation. This is to be expected at or above the flutter speed and illustrates the behavior of the Fourier analysis for a neutrally stable system. As a further demonstration, the same analysis is performed above the flutter speed at 530 ft/s = 6360 in/s, and large dynamic instabilities are apparent in the plotted results as shown in Figure 9-14. The stable mode again damps out and then the unstable mode diverges very rapidly. These rapidly increasing displacements occur as expected when the wing is disturbed above the flutter velocity. The results presented here demonstrate the consistency of NX Nastran between solution sequences (i.e., SOL 145 calculates 5892 in/s = 491 ft/s as the flutter speed and SOL 146 calculates large dynamic responses at or above this value). However, the transient results here must be regarded as more qualitative than quantitative as far as accuracy of a calculated instability boundary (i.e., flutter speed) is concerned.

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Figure 9-12. Wing Tip Displacement Below Flutter Speed

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Figure 9-13. Wing Tip Displacement at Flutter Speed

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Figure 9-14. Wing Tip Displacement Above Flutter Speed N A S T R A N E X E C U T I V E C O N T R O L D E C K E C H O ID NXN, HA146E $$$$$$$$ HANDBOOK FOR AEROELASTIC ANALYSIS EXAMPLE HA146E $$$$$$$$$$$$ $ $ $ MODEL DESCRIPTION MODEL A OF NASA TN D-1824 $ $ HALF SPAN 15 DEGREE SWEPT WING $ $ 28 QUAD4 PANEL MODEL $ $ $ $ SOLUTION TRANSIENT RESPONSE TO AN IMPULSE AT $ $ THE TIP USING DOUBLET LATTICE AERO $ $ $ $ OUTPUT PLOT OF TIP DISPLACEMENT TIME $ $ HISTORY $ $ $ $$$$$$$$ $$$$$$$$ TIME 10 $ CPU TIME IN MINUTES SOL 146 $ AEROELASTIC RESPONSE CEND HALF SPAN 15-DEG SWEPT UNTAPERED WING HA146E PAGE TRANSIENT RESPONSE TO AN IMPULSE, DLM AERO MODEL A OF NASA TN D-1824, 0.45 MACH NO. C A S E C O N T R O L D E C K E C H O CARD COUNT 1 TITLE = HALF SPAN 15-DEG SWEPT UNTAPERED WING HA146E 2 SUBTI = TRANSIENT RESPONSE TO AN IMPULSE, DLM AERO 3 LABEL = MODEL A OF NASA TN D-1824, 0.45 MACH NO. 4 ECHO = BOTH 5 SPC = 1 $ BOUNDARY CONDITIONS 6 METHOD = 10 $ MODIFIED GIVENS EIGENVALUE METHOD 7 SDAMP = 2000 $ STRUCTURAL DAMPING 8 DLOAD = 10 $ LOAD 9 FREQ = 101 $ FREQUENCY LIST 10 TSTEP = 201 $ SOLUTION TIME STEPS

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11 OUTPUT 12 SET 1 = 40 13 DISP = 1 $ DISPLACEMENTS OF POINT 40 14 OUTPUT(XYOUT) 15 CSCALE 2.0 16 PLOTTER = NASTRAN 17 UPPER TICS = -1 18 RIGHT TICS = -1 19 XGRID LINES = YES 20 YGRID LINES = YES 21 XTITLE = TIME (SEC) 22 YTITLE = TIP DISPLACEMENT AT 491 FT/S 23 XYPLOT DISP / 40(T3) 24 BEGIN BULK HALF SPAN 15-DEG SWEPT UNTAPERED WING HA146E TRANSIENT RESPONSE TO AN IMPULSE, DLM AERO MODEL A OF NASA TN D-1824, 0.45 MACH NO. I N P U T B U L K

PAGE D A T A

D E C K

3 E C H O

. 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ $ $ $ THE ANNOTATIONS IN THIS INPUT DECK ARE INTENDED TO $ $ EXPLAIN THE DATA ON THE CARD IMAGES FOR THIS $ $ EXAMPLE WITHOUT REFERENCE TO THE VARIOUS MANUALS WHERE $ $ MORE GENERAL DESCRIPTIONS WILL BE FOUND. $ $ $ $*** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ***$ INCLUDE PLATE_STRUCT.DAT $ $ $ * * MASS AND INERTIA PROPERTIES * * $ $ $ $ THE MAT1 ENTRY DEFINES THE MATERIAL PROPERTIES FOR LINEAR, $ $ TEMPERATURE-INDEPENDENT, ISOTROPIC MATERIALS. IT LISTS $ $ YOUNGS MODULUS, THE SHEAR MODULUS, POISSONS RATIO (NU $ $ DEFAULTS TO .5*E/G-1.0), MASS DENSITY (RHO DEFAULTS TO $ $ ZERO), THERMAL EXPANSION COEFFICIENT (TREF DEFAULTS TO $ $ ZERO) AND STRUCTURAL ELEMENT DAMPING (WHICH IS TWICE THE $ $ CRITICAL DAMPING RATIO AND DEFAULTS TO ZERO). $ $ $ $ MID E G NU RHO A TREF GE MAT1 1 9.2418+63.4993+6 0.097464 $ $ $ THE TABDMP1 ENTRY DEFINES DAMPING COEFFICIENTS VS. FREQUENCY. $ $ VALUES ARE LINEARLY INTERPOLATED AND EXTRAPOLATED. $ $ $ $ ID TYPE TABDMP1 2000 +T2000 $ F1 G1 F2 G2 "ENDT" +T2000 0.0 0.01 1000.0 0.01 ENDT $ $ $ * * STRUCTURAL CONSTRAINTS * * $ $ $ $ THE SPC1 ENTRY DEFINES SETS OF SINGLE-POINT CONSTRAINTS. $ $ IT LISTS THE DOF COMPONENTS TO BE CONSTRAINED AND THE GRID $ $ POINT NO. $ $ $ $ SID C G1 G2 ETC. SPC1 1 6 1 THRU 40 SPC1 1 12345 9 SPC1 1 12345 25 $ $ $ THE ASET1 ENTRY DEFINES DOFS THAT THE USER DESIRES TO BE $ $ INDEPENDENT. IT LISTS THE DOF COMPONENT NUMBERS AND THE $ $ GRID ID NO.S. THIS IS THE ALTERNATE FORM THAT USES THE $ $ ALPHANUMERIC THRU OPTION. $ $ $ $ C G G ETC. ASET1 3 1 THRU 8 ASET1 3 10 THRU 24 ASET1 3 26 THRU 40 $ $ $ * * * AERODYNAMIC DATA * * * $ $ $ $ * * ELEMENT GEOMETRY * * $ $ $ $ THE AERO ENTRY SPECIFIES THE AERO COORDINATE SYSTEM, THE $ $ VELOCITY (USED FOR DATA RECOVERY), THE REFERENCE CHORD $ $ AND FLUID DENSITY, PLUS SYMMETRY KEYS. SYMXZ=1 INDICATES $ $ THAT THE MODEL IS MOUNTED ON A WALL THAT ACTS AS A REFLECTING $ $ PLANE; SYMXY = 0 INDICATES THAT THE MODEL IS MOUNTED FAR $ $ ENOUGH FROM THE FLOOR SO THAT REFLECTION EFFECTS ARE $ $ NEGLIGIBLE. $ $ $ $ ACSID VELOCITY REFC RHOREF SYMXZ SYMXY AERO 0 5892. 2.0706 1.1092-7+1 $ $ $ THE PARAM,Q ENTRY SPECIFIES THE DYNAMIC PRESSURE. (THIS $ $ ENTRY IS REQUIRED IN SOL 146.) $ $ $ PARAM Q 1.925

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$ * * SPLINE FIT ON THE LIFTING SURFACES * * $ $ $ $ * SURFACE SPLINE FIT ON THE WING * $ $ $ THE SPLINE1 ENTRY DEFINES A SURFACE SPLINE FOR INTERPO$ $ LATING OUT-OF-PLANE DISPLACEMENTS FROM THE STRUCTURAL $ $ GRID POINTS ON THE SETG ENTRY TO THE SUB-REGION DEFINED $ $ BY AERODYNAMIC BOXES 101 THRU 124 OF THE REGION ON THE $ $ CAERO1 ENTRY. DZ=0 SPECIFIES THAT NO SMOOTHING OF THE $ $ SPLINE IS TO BE IMPOSED. $ $ $ $ EID CAERO BOX1 BOX2 SETG DZ $ SPLINE1 100 101 101 124 100 .0 $ $ $ THE SET1 ENTRY DEFINES THE SET OF POINTS TO BE USED BY $ $ THE SURFACE SPLINE FOR INTERPOLATION. $ $ $ $ SID G1 G2 G3 G4 ETC SET1 100 2 4 6 8 9 11 13 +S1 +S1 15 18 20 22 24 25 27 29 +S2 +S2 31 34 36 38 40 $ $ $ THE CAERO1 ENTRY IS USED FOR DOUBLET LATTICE AERODYNAMICS. $ $ LISTED ARE ITS PAERO ENTRY ID AND THE COORDINATE SYSTEM $ $ FOR LOCATING THE INBOARD AND OUTBOARD LEADING EDGE POINTS $ $ (1 AND 4). NSPAN AND NCHORD, OR LSPAN AND LCHORD, ARE $ $ USED TO PARTITION THE WING INTO AERODYNAMIC PANELS. $ $ THE FORMER FOR UNIFORMLY SPACED PANELS AND THE LATTER $ $ FOR NON-UNIFORM SPACING. IGID IS THE ID OF ITS ASSOCIATED $ $ INTERFERENCE GROUP. THE CONTINUATION ENTRY DEFINES POINTS $ $ 1 AND 4, THE ROOT CHORD AND TIP CHORD. THE BOXES FORMED $ $ BY THE GRID LINES WILL BE NUMBERED BEGINNING WITH EID $ $ SO CHOOSE A NUMBER THAT IS UNIQUE, AND IS GREATER THAN ALL $ $ STRUCTURAL GRID, SCALAR AND EXTRA POINT IDS. $ $ $ $ EID PID CP NSPAN NCHORD LSPAN LCHORD IGID CAERO1 101 1 0 6 4 1 +CA101 $ ( FWD LEFT POINT ) ROOTCHORD ( FWD RIGHT POINT ) TIP CHORD $ X1 Y1 Z1 X12 X4 Y4 Z4 X14 +CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 $ $ $ THE PAERO1 ENTRY IS REQUIRED EVEN THOUGH IT IS NON-FUNCTIONAL $ $ (BECAUSE THERE ARE NO ASSOCIATED BODIES IN THIS EXAMPLE). $ $ $ $ PID B1 B2 B3 B4 B5 B6 PAERO1 1 $ $ $ $ $ * * * TRANSIENT RESPONSE SOLUTION * * * $ $ $ $ * * AERODYNAMIC CONDITIONS * * $ $ $ $ ALL COMBINATIONS OF MACH NUMBER AND REDUCED FREQUENCY LISTED $ $ ON THE MKAERO1 ENTRY AND ITS CONTINUATION ENTRY WILL BE USED $ $ TO GENERATE GENERALIZED AERO FORCE MATRICES. IF MORE THAN $ $ EIGHT MACH NO.S OR REDUCED FREQUENCIES ARE REQUIRED A SECOND $ $ MKAERO1 ENTRY IS NECESSARY. $ $ $ $ M1 M2 M3 M4 MKAERO1 .45 +MK +MK .001 .10 .12 .14 .16 .20 $ $ $ * * EIGENSOLUTION AND LOAD DATA * * $ $ $ $ THE EIGR ENTRY DEFINES THE METHOD OF EXTRACTING THE EIGEN$ $ VALUES OF THE STRUCTURE. IN THIS CASE THE GIVENS METHOD. $ $ SIX MODES ARE DESIRED, NORMALIZED ON THE MAXIMUM DISPLACEMENTS.$ $ $ $ SID METHOD F1 F2 NE ND EIGR 10 GIV .0 4000. 12 12 +ER $ NORM +ER MAX $ $ $ THE TLOAD1 ENTRY DEFINES A TIME DEPENDENT DYNAMIC LOAD OR AN $ $ ENFORCED MOTION. LISTED ARE THE IDS OF A DAREA ENTRY, A DELAY $ $ ENTRY, THE TYPE OF DYNAMIC EXCITATION AND THE TABLED ID. $ $ $ $ SID DAREA DELAY TYPE TID TLOAD1 10 12 13 14 $ $ THE DAREA ENTRY DEFINES THE POINT WHERE THE DYNAMIC LOAD IS $ $ TO BE APPLIED (IN THIS CASE THE CENTER OF THE SECOND FWDMOST $ $ AERODYNAMIC BOX), THE DOF COMPONENT AND THE ASSOCIATED AREA $ $ SCALE FACTOR. $ $ $ $ SID P C A DAREA 12 24 3 1.0 $ $ $ THE DELAY ENTRY SPECIFIES THE GRID POINT, DISPLACEMENT $ $ COMPONENT AND THE TIME DELAY OF ONSET OF THE FORCING $ $ FUNCTION. $ $ $ $ SID P C T

9-60

NX Nastran Aeroelastic Analysis User’s Guide

Dynamic Aeroelastic Response Analysis

DELAY 13 24 3 .1 $ $ THE TABLED1 ENTRY DEFINES A TABULAR FUNCTION FOR USE IN $ GENERATING FREQUENCY AND TIME DEPENDENT LOADS. IN THIS $ CASE THE PULSE TO BE APPLIED TO THE WING TIP. $ $ SID TABLED1 14 $ X1 Y1 X2 Y2 ETC +T14 0.0 1.+6 1.-6 1.+6 1.-6 -1.+6 2.-6 -1.+6 +T15 ENDT $ $ THE FREQ1 ENTRY DEFINES THE SET OF FREQUENCIES USED TO OBTAIN $ THE FREQUENCY RESPONSE SOLUTION. LISTED ARE THE STARTING $ FREQUENCY, FREQUENCY INCREMENT AND NUMBER OF INCREMENTS. $ $ THE PERIOD OF REPETITION OF THE FORCE IS 1/DF. $ $ SID F1 DF NDF FREQ1 101 0.0 5.0 90 $ $ THE TSTEP ENTRY DEFINES THE TIME STEP INTERVALS AT WHICH THE $ TRANSIENT RESPONSES ARE DESIRED. LISTED ARE THE NUMBER OF $ STEPS, THE TIME INTERVAL AND NO THE INTERVALS FOR OUTPUT. $ $ SID N DT NO TSTEP 201 400 .001 1 $ $ * * PARAMETERS * * $ $ PARAM GRDPNT CAUSES THE INERTIAL MATRIX TO BE PRINTED WITH $ RESPECT TO COORDINATES THROUGH THE GRID POINT SPECIFIED. $ PARAM GRDPNT 17 $ $ THE PARAMETER KDAMP DETERMINES THE MANNER OF INCLUDING $ STRUCTURAL DAMPING IN THE EQUATIONS OF MOTION (SEE THE $ HANDBOOK FOR DYNAMIC ANALYSIS, SECTION 3.2.2). IF SET TO $ -1, MODAL STRUCTURAL DAMPING IS INCLUDED AS THE IMAGINARY $ PART OF A COMPLEX STIFFNESS MATRIX. $ PARAM KDAMP -1 $ $ THE PARAM,LMODES,N ENTRY SPECIFIES THAT N VIBRATION MODES $ ARE TO BE USED IN THE ANALYSIS. $ PARAM LMODES 4 $ ENDDATA INPUT BULK DATA CARD COUNT = 342

NX Nastran Aeroelastic Analysis User’s Guide

$ $ $ $ $ +T14 +T15 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $

9-61

Chapter 9

Dynamic Aeroelastic Response Analysis

Listing 9-16. Input Data Files for Response to Impulsive Loading

9-62

HALF SPAN 15-DEG SWEPT UNTAPERED WING HA146E PAGE 10 TRANSIENT RESPONSE TO AN IMPULSE, DLM AERO MODEL A OF NASA TN D-1824, 0.45 MACH NO. S O R T E D B U L K D A T A E C H O CARD COUNT . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 .. 10 . 1AERO 0 5892. 2.0706 1.1092-7+1 2ASET1 3 1 THRU 8 3ASET1 3 10 THRU 24 4ASET1 3 26 THRU 40 5CAERO1 101 1 0 6 4 1 +CA101 6+CA101 .0 .0 .0 2.07055 1.48044 5.52510 0.0 2.07055 7CQUAD4 1 1 1 2 10 9 +M00000 8+M00000 0.0 0.0 .041 .041 9CQUAD4 2 1 2 3 11 10 +M00001 10+M00001 0.0 0.0 .041 .041 11CQUAD4 3 1 3 4 12 11 +M00002 12+M00002 0.0 0.0 .041 .041 13CQUAD4 4 1 4 5 13 12 +M00003 14+M00003 0.0 0.0 .041 .041 15CQUAD4 5 1 5 6 14 13 +M00004 16+M00004 0.0 0.0 .041 .041 17CQUAD4 6 1 6 7 15 14 +M00005 18+M00005 0.0 0.0 .041 .041 19CQUAD4 7 1 7 8 16 15 +M00006 20+M00006 0.0 0.0 .041 .041 21CQUAD4 8 1 9 10 18 17 22CQUAD4 9 1 10 11 19 18 23CQUAD4 10 1 11 12 20 19 24CQUAD4 11 1 12 13 21 20 25CQUAD4 12 1 13 14 22 21 26CQUAD4 13 1 14 15 23 22 27CQUAD4 14 1 15 16 24 23 28CQUAD4 15 1 17 18 26 25 29CQUAD4 16 1 18 19 27 26 30CQUAD4 17 1 19 20 28 27 31CQUAD4 18 1 20 21 29 28 32CQUAD4 19 1 21 22 30 29 33CQUAD4 20 1 22 23 31 30 34CQUAD4 21 1 23 24 32 31 35CQUAD4 22 1 25 26 34 33 +M00007 36+M00007 .041 .041 0.0 0.0 37CQUAD4 23 1 26 27 35 34 +M00008 38+M00008 .041 .041 0.0 0.0 39CQUAD4 24 1 27 28 36 35 +M00009 40+M00009 .041 .041 0.0 0.0 41CQUAD4 25 1 28 29 37 36 +M00010 42+M00010 .041 .041 0.0 0.0 43CQUAD4 26 1 29 30 38 37 +M00011 44+M00011 .041 .041 0.0 0.0 45CQUAD4 27 1 30 31 39 38 +M00012 46+M00012 .041 .041 0.0 0.0 47CQUAD4 28 1 31 32 40 39 +M00013 48+M00013 .041 .041 0.0 0.0 49DAREA 12 24 3 1.0 50DELAY 13 24 3 .1 51EIGR 10 GIV .0 4000. 12 12 +ER 52+ER MAX 53FREQ1 101 0.0 5.0 90 54GRID 1 0.0 0.0 0.0 55GRID 2 .211491 .7893 0.0 56GRID 3 .422983 1.5786 0.0 57GRID 4 .634474 2.3679 0.0 58GRID 5 .845966 3.1572 0.0 59GRID 6 1.05746 3.9465 0.0 60GRID 7 1.26895 4.7358 0.0 61GRID 8 1.48044 5.5251 0.0 62GRID 9 .258819 0.0 0.0 63GRID 10 .47031 .7893 0.0 64GRID 11 .681802 1.5786 0.0 65GRID 12 .893293 2.3679 0.0 66GRID 13 1.10478 3.1572 0.0 67GRID 14 1.31628 3.9465 0.0 68GRID 15 1.52777 4.7358 0.0 69GRID 16 1.73926 5.5251 0.0 70GRID 17 1.03528 0.0 0.0 71GRID 18 1.24677 .7893 0.0 72GRID 19 1.45826 1.5786 0.0 73GRID 20 1.66975 2.3679 0.0 74GRID 21 1.88124 3.1572 0.0 75GRID 22 2.09273 3.9465 0.0 76GRID 23 2.30422 4.7358 0.0 77GRID 24 2.51572 5.5251 0.0 78GRID 25 1.81173 0.0 0.0 79GRID 26 2.02322 .7893 0.0 80GRID 27 2.23471 1.5786 0.0 81GRID 28 2.44621 2.3679 0.0 82GRID 29 2.6577 3.1572 0.0 83GRID 30 2.86919 3.9465 0.0 84GRID 31 3.08068 4.7358 0.0 85GRID 32 3.29217 5.5251 0.0 NX Nastran Aeroelastic Analysis User’s Guide 86GRID 33 2.07055 0.0 0.0 87GRID 34 2.28204 .7893 0.0 88GRID 35 2.49353 1.5786 0.0

Dynamic Aeroelastic Response Analysis

919293949596979899100101102103104105106107108109110111112113114115116117118-

GRID GRID GRID MAT1 MKAERO1 +MK PAERO1 PARAM PARAM PARAM PARAM PARAM PARAM PSHELL SET1 +S1 +S2 SPC1 SPC1 SPC1 SPLINE1 TABDMP1 +T2000 TABLED1 +T14 +T15 TLOAD1 TSTEP ENDDATA TOTAL COUNT=

38 3.12801 3.9465 39 3.3395 4.7358 40 3.55099 5.5251 1 9.2418+63.4993+6 .45 .001 .10 .12 .14 1 COUPMASS1 GRDPNT 17 KDAMP -1 LMODES 4 Q 1.925 WTMASS .0025901 1 1 .041 1 100 2 4 6 15 18 20 22 31 34 36 38 1 6 1 THRU 1 12345 9 1 12345 25 100 101 101 124 2000 0.0 0.01 1000.0 0.01 14 0.0 1.+6 1.-6 1.+6 ENDT 10 12 13 201 400 .001 1

0.0 0.0 0.0 0.097464 +MK .16

8 24 40 40 100

.20

1 9 25

11 27

13 29

+S1 +S2

.0 +T2000

ENDT 1.-6

-1.+6

2.-6

-1.+6

+T14 +T15

14

119

Listing 9-17. Sorted Bulk Data for Response to Impulsive Loading HALF SPAN 15-DEG SWEPT UNTAPERED WING HA146E TRANSIENT RESPONSE TO AN IMPULSE, DLM AERO MODEL A OF NASA TN D-1824, 0.45 MACH NO. POINT-ID = 40 D I S P L A C E M E N T TIME TYPE 0.0 1.000000E-03 2.000000E-03 3.000000E-03 4.000000E-03 5.000000E-03 6.000001E-03 7.000001E-03 8.000000E-03 9.000001E-03 1.000000E-02 1.100000E-02 1.200000E-02 1.300000E-02 1.400000E-02 1.500000E-02 1.600000E-02 1.700000E-02 1.800000E-02 1.900000E-02 2.000000E-02 2.100000E-02 2.200000E-02 2.300000E-02 2.400000E-02 2.500000E-02 2.600000E-02 2.700000E-02 2.800000E-02 2.900000E-02 3.000000E-02 3.100000E-02 3.200000E-02 3.300000E-02 3.400000E-02 3.500000E-02 3.599999E-02 3.699999E-02 3.799999E-02 3.899999E-02 3.999999E-02 4.099999E-02 4.199998E-02 4.299998E-02 4.399998E-02 4.499998E-02 4.599998E-02 4.699998E-02 4.799997E-02 4.899997E-02 4.999997E-02 5.099997E-02 5.199997E-02 5.299997E-02 5.399996E-02 5.499996E-02

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

T1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

T2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

PAGE

19

V E C T O R

T3 R1 R2 -2.161281E-03 -2.694785E-04 7.338977E-04 -9.106084E-04 -1.683799E-05 4.789137E-04 7.477202E-04 4.718859E-05 -1.609862E-04 2.277885E-03 3.077695E-04 -5.837492E-04 2.869780E-03 5.649505E-04 -8.780780E-04 2.179967E-03 5.142913E-04 -6.107069E-04 2.591812E-04 -6.447673E-05 -3.324048E-04 -1.724737E-03 -3.929759E-04 3.741779E-04 -3.069244E-03 -5.150136E-04 8.119160E-04 -3.035107E-03 -4.234802E-04 1.019383E-03 -1.733900E-03 -3.576560E-04 6.237020E-04 5.516880E-04 3.496103E-05 -3.031057E-05 2.762273E-03 5.524010E-04 -6.235335E-04 3.771174E-03 7.348959E-04 -1.094617E-03 3.087068E-03 5.182477E-04 -9.667448E-04 8.955137E-04 4.657419E-05 -5.671644E-04 -1.699995E-03 -2.580434E-04 3.869796E-04 -3.811964E-03 -6.524998E-04 9.692166E-04 -4.218616E-03 -7.479618E-04 1.320688E-03 -2.774433E-03 -5.563300E-04 9.165197E-04 1.785848E-04 9.601130E-05 2.230951E-04 3.135004E-03 6.180879E-04 -6.944844E-04 4.823445E-03 8.400311E-04 -1.381263E-03 4.398568E-03 7.317043E-04 -1.382091E-03 1.915079E-03 3.196245E-04 -8.582898E-04 -1.566270E-03 -2.366602E-04 2.799646E-04 -4.656748E-03 -8.871090E-04 1.104175E-03 -5.652717E-03 -1.011126E-03 1.723195E-03 -4.183356E-03 -7.404945E-04 1.361757E-03 -5.792427E-04 -1.157179E-05 5.545950E-04 3.378134E-03 5.921518E-04 -7.465787E-04 6.118540E-03 1.079806E-03 -1.669955E-03 6.154180E-03 1.087455E-03 -1.892105E-03 3.357990E-03 5.938875E-04 -1.300285E-03 -1.177767E-03 -2.379042E-04 3.240915E-05 -5.483943E-03 -1.046055E-03 1.253511E-03 -7.413968E-03 -1.284480E-03 2.216528E-03 -6.145379E-03 -1.079321E-03 1.945101E-03 -1.841564E-03 -2.627536E-04 1.013358E-03 3.452360E-03 5.953839E-04 -6.958374E-04 7.620675E-03 1.405553E-03 -1.962146E-03 8.392774E-03 1.481208E-03 -2.554084E-03 5.400310E-03 9.343401E-04 -1.930600E-03 -3.281469E-04 -1.217144E-04 -3.676712E-04 -6.228817E-03 -1.147661E-03 1.381334E-03 -9.583375E-03 -1.677181E-03 2.766101E-03 -8.786102E-03 -1.569629E-03 2.709862E-03 -3.758803E-03 -6.020886E-04 1.679330E-03 3.195483E-03 5.808923E-04 -4.909628E-04 9.246888E-03 1.725094E-03 -2.266037E-03 1.122023E-02 1.952351E-03 -3.378977E-03 8.268386E-03 1.448823E-03 -2.779186E-03 1.187526E-03 1.346227E-04 -9.860625E-04 -6.789724E-03 -1.237243E-03 1.418749E-03 User’s Guide NX Nastran Aeroelastic Analysis -1.216828E-02 -2.180339E-03 3.363178E-03 -1.223997E-02 -2.178860E-03 3.716934E-03

R3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 9-63 0.0 0.0

Chapter 9

5.599996E-02 5.699996E-02 5.799996E-02 5.899996E-02 5.999995E-02 6.099995E-02 6.199995E-02 6.299995E-02 6.399995E-02 6.499995E-02 6.599995E-02 6.699996E-02 6.799996E-02 6.899996E-02 6.999996E-02 7.099997E-02 7.199997E-02 7.299997E-02 7.399997E-02 7.499997E-02 7.599998E-02 7.699998E-02 7.799998E-02 7.899998E-02 7.999998E-02 8.099999E-02 8.199999E-02 8.299999E-02 8.399999E-02 8.499999E-02 8.600000E-02 8.700000E-02 8.800000E-02 8.900000E-02 9.000000E-02 9.100001E-02 9.200001E-02 9.300001E-02 9.400001E-02 9.500001E-02 9.600002E-02 9.700002E-02 9.800002E-02 9.900002E-02 1.000000E-01 1.010000E-01 1.020000E-01 1.030000E-01 1.040000E-01 1.050000E-01 1.060000E-01 1.070000E-01 1.080000E-01 1.090000E-01 1.100000E-01 1.110000E-01 1.120000E-01 1.130001E-01 1.140001E-01 1.150001E-01 1.160001E-01 1.170001E-01 1.180001E-01 1.190001E-01 1.200001E-01 1.210001E-01 1.220001E-01 1.230001E-01 1.240001E-01 1.250001E-01 1.260001E-01 1.270001E-01 1.280001E-01 1.290001E-01 1.300001E-01 1.310001E-01 1.320001E-01 1.330001E-01 1.340001E-01 1.350001E-01 1.360001E-01 1.370001E-01 1.380001E-01 1.390001E-01 1.400001E-01 1.410001E-01 1.420001E-01 1.430001E-01 1.440001E-01 1.450001E-01

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G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

-6.585141E-03 2.365747E-03 1.090876E-02 1.475121E-02 1.219421E-02 3.637522E-03 -6.959493E-03 -1.512759E-02 -1.668699E-02 -1.065213E-02 6.605326E-04 1.245513E-02 1.905753E-02 1.744230E-02 7.401054E-03 -6.423918E-03 -1.837942E-02 -2.232142E-02 -1.635319E-02 -2.322271E-03 1.361746E-02 2.416926E-02 2.433109E-02 1.297798E-02 -4.747977E-03 -2.176206E-02 -2.930862E-02 -2.417410E-02 -7.120336E-03 1.396027E-02 3.008194E-02 3.320365E-02 2.100571E-02 -1.331253E-03 -2.500815E-02 -3.772551E-02 -3.482014E-02 -1.426806E-02 1.261736E-02 3.706274E-02 4.398673E-02 3.280523E-02 4.400965E-03 -2.875855E-02 -1.084636E-02 1.761314E-03 -1.835013E-02 -5.624933E-03 7.094903E-03 -5.870812E-03 -1.496366E-02 -2.143780E-03 1.007576E-02 2.140879E-05 -8.525770E-03 5.925606E-04 9.585598E-03 9.453899E-04 -7.286575E-03 -4.427186E-04 6.698325E-03 3.228778E-04 -6.492403E-03 -5.793792E-04 5.595589E-03 9.382306E-04 -4.914370E-03 -5.407608E-04 4.353250E-03 4.958602E-04 -4.313920E-03 -8.775310E-04 3.668032E-03 8.545416E-04 -2.885876E-03 -4.598846E-04 3.107847E-03 5.193101E-04 -2.859755E-03 -1.054114E-03 2.091984E-03 5.825646E-04 -1.893196E-03 -1.927833E-04 2.239970E-03 8.484185E-04 -1.795201E-03 -8.529953E-04 9.825776E-04 1.680465E-04

-1.086310E-03 4.580293E-04 2.028860E-03 2.570582E-03 2.171182E-03 5.390851E-04 -1.275191E-03 -2.751445E-03 -2.942015E-03 -1.810854E-03 1.788970E-04 2.318320E-03 3.352417E-03 3.119644E-03 1.161095E-03 -1.182372E-03 -3.379039E-03 -3.914595E-03 -2.851486E-03 -3.011227E-04 2.541407E-03 4.292897E-03 4.346912E-03 2.109395E-03 -8.806175E-04 -4.057572E-03 -5.118982E-03 -4.297417E-03 -1.057499E-03 2.589105E-03 5.414958E-03 5.909556E-03 3.507315E-03 -2.684275E-04 -4.769898E-03 -6.508180E-03 -6.369294E-03 -2.054690E-03 2.120463E-03 7.019433E-03 7.507279E-03 5.891309E-03 5.422819E-04 -6.113509E-03 1.394668E-02 6.758023E-04 -3.249122E-02 -6.676541E-04 2.770534E-02 -9.856158E-04 -2.624631E-02 -1.292189E-03 2.327809E-02 7.774846E-04 -2.054968E-02 -1.301968E-03 1.909244E-02 1.525876E-03 -1.674620E-02 -1.687805E-03 1.513168E-02 1.783952E-03 -1.373243E-02 -1.783126E-03 1.215926E-02 2.036112E-03 -1.104605E-02 -1.827426E-03 9.775946E-03 1.906641E-03 -8.901853E-03 -1.918951E-03 7.957551E-03 1.833476E-03 -6.988012E-03 -1.805835E-03 6.451221E-03 1.648978E-03 -5.697040E-03 -1.768444E-03 5.076477E-03 1.579826E-03 -4.532992E-03 -1.422811E-03 4.111860E-03 1.528408E-03 -3.713425E-03 -1.378325E-03 3.126876E-03 1.261815E-03

2.616484E-03 -7.443000E-05 -2.546651E-03 -4.359502E-03 -3.910340E-03 -1.920502E-03 1.295697E-03 3.995620E-03 5.009026E-03 3.894011E-03 6.545364E-04 -2.736080E-03 -5.492608E-03 -5.404990E-03 -3.280462E-03 9.227177E-04 4.615306E-03 6.620240E-03 5.606498E-03 1.838494E-03 -2.745595E-03 -6.765408E-03 -7.340173E-03 -5.202504E-03 1.717783E-04 5.125439E-03 8.594584E-03 7.860237E-03 3.665489E-03 -2.454116E-03 -8.133010E-03 -9.779229E-03 -7.889461E-03 -1.094840E-03 5.329790E-03 1.104246E-02 1.069613E-02 6.447099E-03 -1.718998E-03 -9.542762E-03 -1.259332E-02 -1.202114E-02 -2.173394E-03 3.276326E-03 1.216558E-02 6.118340E-03 -9.178097E-03 1.752827E-03 9.704382E-03 1.912805E-03 -7.775354E-03 -1.487887E-03 7.344068E-03 5.114150E-04 -6.365588E-03 -1.781846E-03 5.963188E-03 8.461776E-04 -5.079691E-03 -1.169393E-03 4.759808E-03 1.309534E-03 -4.258779E-03 -8.945310E-04 3.525509E-03 1.181423E-03 -3.611268E-03 -8.969129E-04 2.912787E-03 1.088643E-03 -2.624643E-03 -8.893022E-04 2.494216E-03 7.230282E-04 -2.110646E-03 -1.050309E-03 1.999515E-03 6.741227E-04 -1.548297E-03 -6.878451E-04 1.651074E-03 7.299435E-04 -1.450750E-03 -6.832129E-04 1.006444E-03 6.216000E-04 -1.192890E-03 -3.824753E-04 1.021087E-03 7.178140E-04

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Dynamic Aeroelastic Response Analysis

1.460001E-01 1.470001E-01 1.480001E-01 1.490001E-01 1.500001E-01 1.510001E-01 1.520001E-01 1.530001E-01 1.540001E-01 1.550001E-01 1.560001E-01 1.570001E-01 1.580001E-01 1.590001E-01 1.600001E-01 1.610001E-01 1.620001E-01 1.630002E-01 1.640002E-01 1.650002E-01 1.660002E-01 1.670002E-01 1.680002E-01 1.690002E-01 1.700002E-01 1.710002E-01 1.720002E-01 1.730002E-01 1.740002E-01 1.750002E-01 1.760002E-01 1.770002E-01 1.780002E-01 1.790002E-01 1.800002E-01 1.810002E-01 1.820002E-01 1.830002E-01 1.840002E-01 1.850002E-01 1.860002E-01 1.870002E-01 1.880002E-01 1.890002E-01 1.900002E-01 1.910002E-01 1.920002E-01 1.930002E-01 1.940002E-01 1.950002E-01 1.960002E-01 1.970002E-01 1.980002E-01 1.990002E-01 2.000002E-01 2.010002E-01 2.020002E-01 2.030002E-01 2.040002E-01 2.050002E-01 2.060002E-01 2.070002E-01 2.080002E-01 2.090002E-01 2.100002E-01 2.110002E-01 2.120003E-01 2.130003E-01 2.140003E-01 2.150003E-01 2.160003E-01 2.170003E-01 2.180003E-01 2.190003E-01 2.200003E-01 2.210003E-01 2.220003E-01 2.230003E-01 2.240003E-01 2.250003E-01 2.260003E-01 2.270003E-01 2.280003E-01 2.290003E-01 2.300003E-01 2.310003E-01 2.320003E-01 2.330003E-01 2.340003E-01 2.350003E-01 2.360003E-01 2.370003E-01 2.380003E-01

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-1.493984E-03 -2.696360E-04 1.684244E-03 1.002875E-03 -6.957233E-04 -4.900639E-04 6.023014E-04 -2.351499E-04 -1.380059E-03 -6.787569E-04 9.882766E-04 9.362240E-04 5.189726E-05 1.955679E-04 6.786224E-04 -8.227860E-05 -1.345931E-03 -1.051940E-03 1.071181E-05 4.550409E-04 2.333840E-04 7.116988E-04 1.153325E-03 4.913089E-04 -7.658474E-04 -1.200295E-03 -7.489835E-04 -5.185438E-04 -2.042031E-04 5.615253E-04 1.487935E-03 1.273256E-03 3.024940E-04 -5.688225E-04 -9.792754E-04 -1.274795E-03 -1.241922E-03 -2.804317E-04 9.704945E-04 1.702863E-03 1.403530E-03 7.665304E-04 -1.959836E-04 -1.251860E-03 -1.988690E-03 -1.660003E-03 -3.841232E-04 9.434439E-04 1.863529E-03 2.019248E-03 1.474028E-03 -1.839087E-05 -1.600760E-03 -2.515950E-03 -2.160876E-03 -9.102881E-04 7.480578E-04 2.278186E-03 2.869659E-03 2.179565E-03 2.587293E-04 -1.725127E-03 -3.069415E-03 -3.034864E-03 -1.733352E-03 5.522765E-04 2.762739E-03 3.771152E-03 3.086632E-03 8.947147E-04 -1.700743E-03 -3.812307E-03 -4.218458E-03 -2.773780E-03 1.795202E-04 3.135700E-03 4.823594E-03 4.398130E-03 1.914227E-03 -1.567403E-03 -4.657363E-03 -5.652663E-03 -4.182603E-03 -5.780478E-04 3.378979E-03 6.118981E-03 6.153750E-03 3.356797E-03 -1.179238E-03 -5.484900E-03 -7.414041E-03 -6.144434E-03 -1.839957E-03

-2.958461E-03 -1.161745E-03 2.685519E-03 1.226687E-03 -2.232762E-03 -1.113328E-03 2.037505E-03 8.511588E-04 -1.900070E-03 -1.018426E-03 1.711110E-03 9.626952E-04 -1.327085E-03 -7.185966E-04 1.298382E-03 7.057204E-04 -1.331449E-03 -8.056768E-04 9.181313E-04 7.198871E-04 -8.213215E-04 -4.137290E-04 9.373685E-04 6.023816E-04 -7.955685E-04 -7.098395E-04 4.777487E-04 3.266047E-04 -5.216984E-04 -3.326587E-04 7.536719E-04 5.774245E-04 -3.458402E-04 -4.596818E-04 1.816574E-04 1.023419E-04 -5.531911E-04 -3.115422E-04 4.340205E-04 6.021237E-04 -2.958229E-05 -9.373976E-05 1.585177E-04 8.339703E-06 -5.505313E-04 -5.022489E-04 1.249955E-04 3.465624E-04 2.101642E-04 1.558693E-04 4.132012E-04 1.189617E-04 -3.883835E-04 -6.093766E-04 -2.693361E-04 -1.687789E-05 4.728562E-05 3.079034E-04 5.649793E-04 5.141765E-04 -6.459943E-05 -3.929432E-04 -5.150104E-04 -4.234743E-04 -3.575768E-04 3.508538E-05 5.524859E-04 7.348440E-04 5.181605E-04 4.645199E-05 -2.581777E-04 -6.525432E-04 -7.479748E-04 -5.561418E-04 9.624101E-05 6.181641E-04 8.400299E-04 7.316995E-04 3.195525E-04 -2.369589E-04 -8.873062E-04 -1.011140E-03 -7.403171E-04 -1.130328E-05 5.921744E-04 1.079928E-03 1.087471E-03 5.936244E-04 -2.381812E-04 -1.046208E-03 -1.284453E-03 -1.079112E-03 -2.625199E-04

-8.348633E-04 -5.194459E-04 6.588913E-04 2.470292E-04 -7.416279E-04 -5.288565E-04 7.820815E-04 4.781105E-04 -2.816594E-04 -3.198939E-04 4.971909E-04 1.674076E-04 -6.418804E-04 -4.638136E-04 2.630581E-04 4.385834E-04 -1.954969E-04 5.613884E-05 3.878956E-04 3.102073E-04 -4.760102E-04 -4.676717E-04 -3.286889E-05 6.817585E-05 -9.293895E-05 1.382808E-05 5.672128E-04 3.524377E-04 -3.804082E-05 -4.009234E-04 -1.498162E-04 -2.459310E-04 -3.141850E-04 -8.242772E-05 3.824324E-04 5.850311E-04 2.107370E-04 7.864104E-05 -1.594367E-04 -2.549823E-04 -6.121827E-04 -3.773403E-04 3.192795E-05 4.258728E-04 4.967770E-04 4.024686E-04 3.376834E-04 -1.410667E-04 -4.862250E-04 -7.712894E-04 -3.943830E-04 -9.099875E-05 3.585487E-04 6.101881E-04 7.338890E-04 4.787360E-04 -1.610488E-04 -5.838384E-04 -8.780280E-04 -6.106258E-04 -3.323081E-04 3.743645E-04 8.119692E-04 1.019369E-03 6.235040E-04 -3.043863E-05 -6.237298E-04 -1.094646E-03 -9.666639E-04 -5.669380E-04 3.872023E-04 9.693527E-04 1.320671E-03 9.163487E-04 2.228875E-04 -6.947663E-04 -1.381313E-03 -1.382021E-03 -8.579714E-04 2.802343E-04 1.104366E-03 1.723242E-03 1.361565E-03 5.543356E-04 -7.469959E-04 -1.670057E-03 -1.892052E-03 -1.299945E-03 3.281619E-05 1.253870E-03 2.216629E-03 1.944911E-03 1.012914E-03

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

9-65

Chapter 9

2.390003E-01 2.400003E-01 2.410003E-01 2.420003E-01 2.430003E-01 2.440003E-01 2.450003E-01 2.460003E-01 2.470003E-01 2.480003E-01 2.490003E-01 2.500003E-01 2.510003E-01 2.520003E-01 2.530003E-01 2.540003E-01 2.550003E-01 2.560003E-01 2.570002E-01 2.580002E-01 2.590002E-01 2.600002E-01 2.610002E-01 2.620002E-01 2.630002E-01 2.640001E-01 2.650001E-01 2.660001E-01 2.670001E-01 2.680001E-01 2.690001E-01 2.700001E-01 2.710001E-01 2.720000E-01 2.730000E-01 2.740000E-01 2.750000E-01 2.760000E-01 2.770000E-01 2.780000E-01 2.790000E-01 2.799999E-01 2.809999E-01 2.819999E-01 2.829999E-01 2.839999E-01 2.849999E-01 2.859999E-01 2.869999E-01 2.879998E-01 2.889998E-01 2.899998E-01 2.909998E-01 2.919998E-01 2.929998E-01 2.939998E-01 2.949997E-01 2.959997E-01 2.969997E-01 2.979997E-01 2.989997E-01 2.999997E-01 3.009997E-01 3.019997E-01 3.029996E-01 3.039996E-01 3.049996E-01 3.059996E-01 3.069996E-01 3.079996E-01 3.089996E-01 3.099996E-01 3.109995E-01 3.119995E-01 3.129995E-01 3.139995E-01 3.149995E-01 3.159995E-01 3.169995E-01 3.179995E-01 3.189994E-01 3.199994E-01 3.209994E-01 3.219994E-01 3.229994E-01 3.239994E-01 3.249994E-01 3.259993E-01 3.269993E-01 3.279993E-01 3.289993E-01

9-66

Dynamic Aeroelastic Response Analysis

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NX Nastran Aeroelastic Analysis User’s Guide

3.453881E-03 7.621572E-03 8.392469E-03 5.398798E-03 -3.302165E-04 -6.230413E-03 -9.583949E-03 -8.785130E-03 -3.756500E-03 3.197917E-03 9.248429E-03 1.122016E-02 8.266538E-03 1.184665E-03 -6.792149E-03 -1.216930E-02 -1.223911E-02 -6.582852E-03 2.368485E-03 1.091072E-02 1.475148E-02 1.219281E-02 3.635037E-03 -6.961936E-03 -1.512885E-02 -1.668653E-02 -1.065038E-02 6.627294E-04 1.245672E-02 1.905787E-02 1.744150E-02 7.399579E-03 -6.425262E-03 -1.838016E-02 -2.232134E-02 -1.635271E-02 -2.321702E-03 1.361780E-02 2.416934E-02 2.433116E-02 1.297826E-02 -4.747333E-03 -2.176146E-02 -2.930844E-02 -2.417504E-02 -7.122335E-03 1.395812E-02 3.008059E-02 3.320433E-02 2.100846E-02 -1.327078E-03 -2.500460E-02 -3.772444E-02 -3.482290E-02 -1.427403E-02 1.261101E-02 3.705790E-02 4.398796E-02 3.281017E-02 4.413120E-03 -2.875510E-02 -1.085742E-02 1.766391E-03 -1.834716E-02 -5.634046E-03 7.096197E-03 -5.864178E-03 -1.496431E-02 -2.152263E-03 1.007565E-02 2.832067E-05 -8.526123E-03 5.856552E-04 9.585500E-03 9.524014E-04 -7.286367E-03 -4.487576E-04 6.697942E-03 3.291314E-04 -6.492182E-03 -5.850392E-04 5.594981E-03 9.434395E-04 -4.913731E-03 -5.457957E-04 4.353001E-03 5.005336E-04 -4.313071E-03 -8.822255E-04 3.667275E-03 8.584516E-04

5.956089E-04 1.405847E-03 1.481200E-03 9.340101E-04 -1.220906E-04 -1.147759E-03 -1.677284E-03 -1.569489E-03 -6.016875E-04 5.812791E-04 1.725327E-03 1.952431E-03 1.448477E-03 1.339732E-04 -1.237627E-03 -2.180480E-03 -2.178696E-03 -1.086005E-03 4.586253E-04 2.029289E-03 2.570610E-03 2.170888E-03 5.386329E-04 -1.275574E-03 -2.751689E-03 -2.941853E-03 -1.810635E-03 1.793185E-04 2.318546E-03 3.352434E-03 3.119428E-03 1.160860E-03 -1.182623E-03 -3.379212E-03 -3.914532E-03 -2.851291E-03 -3.010529E-04 2.541347E-03 4.292966E-03 4.347008E-03 2.109392E-03 -8.806002E-04 -4.057511E-03 -5.118906E-03 -4.297596E-03 -1.057946E-03 2.588708E-03 5.414738E-03 5.909710E-03 3.507734E-03 -2.676054E-04 -4.769334E-03 -6.507999E-03 -6.369889E-03 -2.055919E-03 2.119660E-03 7.018307E-03 7.507920E-03 5.891366E-03 5.459607E-04 -6.116121E-03 1.394087E-02 6.902323E-04 -3.249004E-02 -6.863975E-04 2.770672E-02 -9.690460E-04 -2.624627E-02 -1.309318E-03 2.327812E-02 7.933610E-04 -2.054936E-02 -1.317151E-03 1.909170E-02 1.540756E-03 -1.674549E-02 -1.701517E-03 1.513037E-02 1.797356E-03 -1.373120E-02 -1.795491E-03 1.215771E-02 2.047807E-03 -1.104445E-02 -1.838428E-03 9.774495E-03 1.917038E-03 -8.899920E-03 -1.928860E-03 7.955862E-03 1.842394E-03

-6.963862E-04 -1.962363E-03 -2.554140E-03 -1.930166E-03 -3.671448E-04 1.382003E-03 2.766317E-03 2.709697E-03 1.678723E-03 -4.917661E-04 -2.266577E-03 -3.379125E-03 -2.778684E-03 -9.854110E-04 1.419625E-03 3.363552E-03 3.716870E-03 2.615821E-03 -7.517185E-05 -2.547267E-03 -4.359750E-03 -3.910022E-03 -1.919886E-03 1.296542E-03 3.996010E-03 5.009115E-03 3.893492E-03 6.539468E-04 -2.736659E-03 -5.492821E-03 -5.404855E-03 -3.280084E-03 9.231617E-04 4.615506E-03 6.620323E-03 5.606419E-03 1.838321E-03 -2.745761E-03 -6.765412E-03 -7.340153E-03 -5.202598E-03 1.715333E-04 5.125216E-03 8.594471E-03 7.860482E-03 3.665930E-03 -2.453377E-03 -8.132558E-03 -9.779184E-03 -7.890218E-03 -1.096041E-03 5.328638E-03 1.104175E-02 1.069685E-02 6.448292E-03 -1.716477E-03 -9.541666E-03 -1.259255E-02 -1.202309E-02 -2.176007E-03 3.274524E-03 1.216307E-02 6.124929E-03 -9.177285E-03 1.746809E-03 9.704980E-03 1.917567E-03 -7.773848E-03 -1.493671E-03 7.343920E-03 5.161318E-04 -6.364749E-03 -1.786556E-03 5.962196E-03 8.510052E-04 -5.079296E-03 -1.173318E-03 4.758496E-03 1.313984E-03 -4.258122E-03 -8.981298E-04 3.524496E-03 1.185068E-03 -3.610261E-03 -9.004638E-04 2.912089E-03 1.091568E-03 -2.623410E-03 -8.925656E-04 2.493622E-03 7.256354E-04

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Dynamic Aeroelastic Response Analysis

3.299993E-01 3.309993E-01 3.319993E-01 3.329993E-01 3.339992E-01 3.349992E-01 3.359992E-01 3.369992E-01 3.379992E-01 3.389992E-01 3.399992E-01 3.409992E-01 3.419991E-01 3.429991E-01 3.439991E-01 3.449991E-01 3.459991E-01 3.469991E-01 3.479991E-01 3.489991E-01 3.499990E-01 3.509990E-01 3.519990E-01 3.529990E-01 3.539990E-01 3.549990E-01 3.559990E-01 3.569990E-01 3.579989E-01 3.589989E-01 3.599989E-01 3.609989E-01 3.619989E-01 3.629989E-01 3.639989E-01 3.649988E-01 3.659988E-01 3.669988E-01 3.679988E-01 3.689988E-01 3.699988E-01 3.709988E-01 3.719988E-01 3.729987E-01 3.739987E-01 3.749987E-01 3.759987E-01 3.769987E-01 3.779987E-01 3.789987E-01 3.799987E-01 3.809986E-01 3.819986E-01 3.829986E-01 3.839986E-01 3.849986E-01 3.859986E-01 3.869986E-01 3.879986E-01 3.889985E-01 3.899985E-01 3.909985E-01 3.919985E-01 3.929985E-01 3.939985E-01 3.949985E-01 3.959984E-01 3.969984E-01 3.979984E-01 3.989984E-01 3.999984E-01

G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-2.885107E-03 -4.639394E-04 3.107272E-03 5.230833E-04 -2.858792E-03 -1.057238E-03 2.090989E-03 5.855286E-04 -1.893007E-03 -1.954876E-04 2.239133E-03 8.513575E-04 -1.794120E-03 -8.552190E-04 9.818454E-04 1.700623E-04 -1.493627E-03 -2.722920E-04 1.683268E-03 1.004625E-03 -6.945509E-04 -4.914580E-04 6.020381E-04 -2.334317E-04 -1.379630E-03 -6.807937E-04 9.870963E-04 9.373573E-04 5.249940E-05 1.949151E-04 6.786276E-04 -8.058648E-05 -1.345312E-03 -1.053175E-03 9.650893E-06 4.549030E-04 2.334058E-04 7.107151E-04 1.153458E-03 4.928737E-04 -7.643981E-04 -1.200567E-03 -7.495407E-04 -5.188594E-04 -2.048586E-04 5.599606E-04 1.487058E-03 1.274628E-03 3.040848E-04 -5.678338E-04 -9.789682E-04 -1.274341E-03 -1.242691E-03 -2.823023E-04 9.684587E-04 1.702390E-03 1.404470E-03 7.678129E-04 -1.939435E-04 -1.250422E-03 -1.988131E-03 -1.661686E-03 -3.864922E-04 9.413488E-04 1.862532E-03 2.019427E-03 1.475807E-03 -1.539207E-05 -1.598186E-03 -2.515437E-03 -2.162746E-03

-6.986035E-03 -1.814380E-03 6.449440E-03 1.656828E-03 -5.695171E-03 -1.775701E-03 5.074440E-03 1.586703E-03 -4.531471E-03 -1.428696E-03 4.109832E-03 1.534250E-03 -3.711471E-03 -1.383589E-03 3.125043E-03 1.266645E-03 -2.956689E-03 -1.166522E-03 2.683830E-03 1.230594E-03 -2.230736E-03 -1.117147E-03 2.035938E-03 8.546368E-04 -1.898591E-03 -1.021672E-03 1.709569E-03 9.657767E-04 -1.325802E-03 -7.211514E-04 1.297060E-03 7.082741E-04 -1.330153E-03 -8.078989E-04 9.169857E-04 7.216685E-04 -8.201152E-04 -4.156126E-04 9.364379E-04 6.041599E-04 -7.941440E-04 -7.111878E-04 4.767647E-04 3.275825E-04 -5.209335E-04 -3.341018E-04 7.525436E-04 5.788830E-04 -3.448114E-04 -4.606239E-04 1.807473E-04 1.033490E-04 -5.525769E-04 -3.125830E-04 4.329739E-04 6.026756E-04 -2.862414E-05 -9.405064E-05 1.582681E-04 8.981047E-06 -5.497173E-04 -5.030980E-04 1.240874E-04 3.463441E-04 2.104558E-04 1.554509E-04 4.130549E-04 1.199204E-04 -3.876524E-04 -6.096095E-04 -2.701208E-04

-2.109552E-03 -1.052916E-03 1.998578E-03 6.765602E-04 -1.547721E-03 -6.898907E-04 1.649966E-03 7.323427E-04 -1.450125E-03 -6.847495E-04 1.005443E-03 6.233312E-04 -1.192173E-03 -3.842200E-04 1.020451E-03 7.190913E-04 -8.337529E-04 -5.209653E-04 6.585512E-04 2.479874E-04 -7.407800E-04 -5.302284E-04 7.813104E-04 4.791183E-04 -2.811395E-04 -3.205045E-04 4.965789E-04 1.686769E-04 -6.416096E-04 -4.644895E-04 2.621408E-04 4.392531E-04 -1.951410E-04 5.552897E-05 3.877376E-04 3.109371E-04 -4.751335E-04 -4.683674E-04 -3.314081E-05 6.816424E-05 -9.255492E-05 1.309348E-05 5.669591E-04 3.529138E-04 -3.739426E-05 -4.008568E-04 -1.501879E-04 -2.454688E-04 -3.144755E-04 -8.296494E-05 3.815530E-04 5.854531E-04 2.111366E-04 7.892682E-05 -1.591782E-04 -2.545648E-04 -6.118199E-04 -3.781085E-04 3.128907E-05 4.252739E-04 4.971457E-04 4.023444E-04 3.383066E-04 -1.404991E-04 -4.855430E-04 -7.713680E-04 -3.951788E-04 -9.146095E-05 3.576092E-04 6.100373E-04 7.337374E-04

NX Nastran Aeroelastic Analysis User’s Guide

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

9-67

Chapter 9

Dynamic Aeroelastic Response Analysis

Listing 9-18. Output for Response to Impulsive Loading

9-68

NX Nastran Aeroelastic Analysis User’s Guide

Appendix

A

References and Bibliography

This appendix includes references of interest in the field of aeroelastic analysis and design. Three categories are included. The first category, References, contains the literature cited directly in this guide. The second category, Bibliography of Aeroelasticity and Unsteady Aerodynamics, contains textbook references to the fundamentals of aerodynamic/structural interaction. The third category, Bibliography of MSC.Nastran Applications, has been excerpted from the MSC.Nastran Bibliography and includes specific solutions to problems in aeroelasticity by the methods described in this guide.

A.1

References

The references below are all cited in the text of this guide. These are listed alphabetically by the first author. In the case of multiple references to the same author, the references are listed chronologically by year. For multiple references to the same author in the same year, letters in alphabetical order are appended to the year. Albano, E., and Rodden, W. P. (1969) . A Doublet-Lattice Method for Calculating Lift Distributions on Oscillating Surfaces in Subsonic Flows. AIAA J. , Vol. 7, pp. 279-285, p. 2192. Anon. (1961) . Proceedings of Symposium on Aerothermoelasticity. Air Force Systems Command, ASD Technical Report 61-645. Anon. (1971) . “Modal Flutter Analysis Study; Vol. IV, Aero-Servo-Elastic Stability Analysis Program: PASES,” Rpt. No. SD15204R, Hughes Aircraft Company. Ashley, H., and Zartarian, G. (1956) . Piston Theory - A New Aerodynamic Tool for the Aeroelastician. J. Aero. Sci. , Vol. 23, pp. 1109-1118. Bisplinghoff, R. L. (1956) . Some Structural and Aeroelastic Considerations of High Speed Flight. J. Aero. Sci ., Vol. 23, pp. 289-329, 367. Bisplinghoff, R. L., Ashley, H., and Halfman, R. L. (1955) . Aeroelasticity . Reading: Addison-Wesley Publishing Company. Blakely, K. (1993) . MSC/NASTRAN Basic Dynamic Analysis User’s Guide , Version 68. The MacNeal-Schwendler Corporation. Bryan, G. H. (1911) . Stability in Aviation . London: MacMillan Company. Budiansky, B., and Mayers, J. (1956) . Influence of Aerodynamic Heating on the Effective Torsional Stiffness of Thin Wings. J. Aero. Sci ., Vol. 23, pp. 1081-1093, 1108. Caffrey, J. P., and Lee, J. M. (1994) . MSC/NASTRAN Linear Static Analysis User’s Guide , Version 68. The MacNeal-Schwendler Corporation. Chen, P. C., and Liu, D. D. (1985) . A Harmonic Gradient Method for Unsteady Supersonic Flow Calculations. J. Aircraft , Vol. 22, pp. 371-379.

NX Nastran Aeroelastic Analysis User’s Guide

A-1

Appendix A

References and Bibliography

Desmarais, R. N., and Bennett, R. M. (1974) . An Automated Procedure for Computing Flutter Eigenvalues. J. Aircraft , Vol. 11, pp. 75-80. Donato, V. W., and Huhn, C. R., Jr. (1968) . Supersonic Unsteady Aerodynamics for Wings with Trailing-edge Control Surfaces and Folded Tips. Air Force Flight Dynamics Laboratory Report No. AFFDL-TR-68-30. Etkin, B. (1972) . Dynamics of Atmospheric Flight . New York: John Wiley & Sons. Finck, R. D., and Hoak, D. E. (1976) . USAF Stability and Control DATCOM. Air Force Flight Dynamics Laboratory, Flight Controls Division. Frazer, R. A., and Duncan, W. J. (1928) . The Flutter of Aeroplane Wings. British A.R.C., R. & M. 1155. Friedman, I.P., Vosteen, L.F., Cooper, M.J., Gold, R.R., and Reed, W.H.,III (1988) . Assessment of Thermoelastic Analysis and Testing Applicable to the National Aero-Space Plane. NASP Contractor Report 1017. Giesing, J. P., Kalman, T. P., and Rodden, W. P. (1971) . Subsonic Unsteady Aerodynamics for General Configurations; Part I, Vol. I - Direct Application of the Nonplanar Doublet-Lattice Method. Air Force Flight Dynamics Laboratory Report No. AFFDL-TR-71-5, Part I, Vol. I. Giesing, J. P., Kalman, T., and Rodden, W. P. (1972a) . Subsonic Unsteady Aerodynamics for General Configurations, Part II, Volume I - Application of the Doublet-Lattice-Method and the Method of Images to Lifting-Surface/Body Interference. Air Force Flight Dynamics Laboratory Report No. AFFDL-TR-71-5, Part II, Vol. I. Giesing, J. P., Kalman, T. P., and Rodden, W. P. (1972b) . Subsonic Unsteady Aerodynamics for General Configurations; Part II, Volume II - Computer Program N5KA. Air Force Flight Dynamics Laboratory Report No. AFFDL-TR-71-5, Part II, Vol. II. Giesing, J. P., Kalman, T. P., and Rodden, W. P. (1972c) . Subsonic Steady and Oscillatory Aerodynamics for Multiple Interfering Wings and Bodies. J. Aircraft , Vol. 9, pp. 693-702. Giesing, J. P., Kalman, T. P., and Rodden, W. P. (1976) . Correction Factor Techniques for Improving Aerodynamic Prediction Methods. NASA CR 144967, May 1976. Gockel, M. A. (1983) . MSC/NASTRAN Handbook for Dynamic Analysis , Version 63. The MacNeal-Schwendler Corporation. Harder, R. L., and Desmarais, R. N. (1972a) . Interpolation Using Surface Splines. J. Aircraft , Vol. 9, pp. 189-191. Harder, R. L, and Desmarais, R. N. (1972b) . Reply by Authors to W. P. Rodden, J. A. McGrew, and T. P. Kalman. J. Aircraft , Vol. 9, p. 871. Harder, R. L., MacNeal, R. H., and Rodden, W. P. (1971) . A Design Study for the Incorporation of Aeroelastic Capability into NASTRAN. NASA CR-111918. Hassig, H. J. (1971) . An Approximate True Damping Solution of the Flutter Equation by Determinant Iteration. J. Aircraft . Vol. 8, pp. 885-889. Hedgepeth, J. M. (1957) . Flutter of Rectangular Simply Supported Panels at High Supersonic Speeds. J. Aero. Sci. , Vol. 24, pp. 563-573, 586. Johnson, E. H., and Venkayya, V. B., (1988). Automated Structural Optimization System (ASTROS), Volume 1. Theoretical Manual, AFWAL-TR-88,3028, Volume 1. Johnson, E. H., and Reymond, M. A., Multidisciplinary Aeroelastic Analysis and Design Using MSC/NASTRAN (1991) . Proceedings of the AIAA/ASME/ASCE/AHS/ASG 32nd Structures, Structural Dynamics, and Materials Conference AIAA-91-1097-CP, Baltimore, MD.

A-2

NX Nastran Aeroelastic Analysis User’s Guide

References and Bibliography

Lahey, R. S., Miller, M. P. and Reymond, M. (1994) . MSC/NASTRAN Reference Manual , Version 68. The MacNeal-Schwendler Corporation. Lawrence, A. J., and Jackson, P. (1970) . Comparison of Different Methods of Assessing the Free Oscillatory Characteristics of Aeroelastic Systems. British A.R.C. C.P. No. 1084. Komzsik, L. (1993) MSC/NASTRAN Numerical Methods User’s Guide . The MacNeal-Schwendler Corporation. Küssner, H. G., and Schwarz, I. (1940) . Der schwingenden Flugel mit aerodynamisch ausgeglichenem Ruder. Luftfahrtforschung, Vol. 17, pp. 337-354; translated as The Oscillating Wing with Aerodynamically Balanced Elevator, NACA TM-991. Liu, D. D., James, D. K., Chen, P. C., and Pototzky, A. S. (1991) . Further Studies of Harmonic Gradient Method for Supersonic Aeroelastic Applications. J. Aircraft , Vol. 28, pp. 598-605. Moore, G. J. (1994) . MSC/NASTRAN Design Sensitivity and Optimization User’s Guide , Version 68. The MacNeal-Schwendler Corporation. Moore, M. T., and Andrew, L. V. (1965) . Unsteady Aerodynamics for Advanced Configurations, Part IV - Application of the Supersonic Mach Box Method to Intersecting Planar Lifting Surfaces. Air Force Flight Dynamics Laboratory Report, No. FDL-TDR-64-152, Part IV. Neill, D. N., Johnson, E. H., and Canfield, R. A., (1987). ASTROS - A Multidisciplinary Automated Structural Design Tool. Proceedings of the AIAA/ASME/ASCE/AHS/ASC 28th Structures, Structural Dynamics and Materials Conference. AIAA-87-0713-CP, Monterey, California. Pendleton, E. W., French, M., and Noll, T. E. (1987) . A Comparison of Flutter Analyses for a 45° Swept Transonic Model. Proceedings of the AIAA Design, Systems & Operations Meeting, St. Louis, MO. Pines, S., Dugundji, J., and Neuringer, J. (1955) . Aerodynamic Flutter Derivatives for a Flexible Wing with Supersonic and Subsonic Edges. J. Aero. Sci. , Vol. 22, pp. 693-700. Reymond, M. and Miller, M. P. (1994) . MSC/NASTRAN Quick Reference Guide , Version 68. The MacNeal-Schwendler Corporation. Rodden, W. P. (1959a) . A Matrix Approach to Flutter Analysis. Institute of the Aeronautical Sciences, Fairchild Fund Paper No. FF-23. Rodden, W. P. (1959b) . Further Remarks on “Matrix Interpolation of Flexibility Influence Coefficients.” J. Aero. Sci. , Vol. 13, p. 760. Rodden, W. P. (1977) . Comment on “Flutter of Flat Finite Element Panels in a Supersonic Potential Flow.” AIAA J. , Vol. 15, pp. 755-756. Rodden, W. P. (ed.) (1987) . MSC/NASTRAN Handbook for Aeroelastic Analysis. The MacNealSchwendler Corp., Report MSR-57. Rodden, W. P., and Bellinger, E. D. (1982a) . Aerodynamic Lag Functions, Divergence, and the British Flutter Method. J. Aircraft , Vol. 19, pp. 596-598. Rodden, W. P., and Bellinger, E. D. (1982b) . Unrestrained Aeroelastic Divergence in a Dynamic Stability Analysis. J. Aircraft , Vol. 19, pp. 796-797. Rodden, W. P., Bellinger, E. D., and Giesing, J. P. (1984) . Errata and Addenda to “Application of Oscillatory Aerodynamic Theory to Estimation of Dynamic Stability Derivatives.” J. Aircraft , Vol. 21, pp. 93-94. Rodden, W. P., Farkas, E. F., Malcom, H. A., and Kliszewski, A. M. (1962) . Aerodynamic Influence Coefficients from Piston Theory: Analytical Development and Computational Procedure. Aerospace Corp., Report No. TDR-169(3230-11)TN-2.

NX Nastran Aeroelastic Analysis User’s Guide

A-3

Appendix A

References and Bibliography

Rodden, W. P., and Giesing, J. P. (1970) . Application of Oscillatory Aerodynamic Theory to Estimation of Dynamic Stability Derivatives. J. Aircraft , Vol. 7, pp. 272-275. Rodden, W. P., Giesing, J. P., and Kalman, T. P. (1972) . Refinement of the Nonplanar Aspects of the Subsonic Doublet-Lattice Lifting Surface Method. J. Aircraft , Vol. 9, pp. 69-73. Rodden, W. P., Harder, R. L., and Bellinger, E. D. (1979) . Aeroelastic Addition to NASTRAN. NASA Contractor Report 3094. Rodden, W. P., and Love, J. R. (1985) . Equations of Motion of a Quasisteady Flight Vehicle Utilizing Restrained Static Aeroelastic Characteristics. J. Aircraft , Vol. 22, pp. 802-809. Rodden W. P., McGrew, J. A., and Kalman, T.P. (1972) . Comment on “Interpolation Using Surface Splines” J. Aircraft , Vol. 9, pp. 869-871. Rodden, W. P. and Harder, R. L. (1975). Flutter Analysis with Active Controls, Proceedings of MSC/NASTRAN Users’ Conference, Pasadena, CA. Rodden, W. P., and Revell, J. D. (1962) . The Status of Unsteady Aerodynamic Influence Coefficients. Institute of the Aeronautical Sciences, Fairchild Fund Paper No. FF-33. Schmitt, A. F. (1956) . A Least Squares Matrix Interpolation of Flexibility Influence Coefficients. J. Aero. Sci. , Vol. 10, p. 980. Smilg, B., and Wasserman, L. S. (1942) . Application of Three-Dimensional Flutter Theory to Aircraft Structures. Army Air Force Tech. Rpt. 4798. Taylor, J. (ed.). (1965) . Manual on Aircraft Loads . AGARDograph 83, Pergamon Press. Theodorsen, T. (1935) . General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA Report 496. Tuovila, W.J., and McCarty, J. L., (1955) Experimental Flutter Results for Cantilever Wing Models at Mach Numbers Up to 3.0, NACA RM L55E11. Turner, M.J., Dill, E.H., Martin, H.C., and Melosh, R.J. (1960) . Large Deflections of Structures Subjected to Heating and External Loads. J. Aero/Space Sci ., Vol. 27, pp. 97-106, 127. Vanderplaats, G. N. (1984) . Numerical Optimization Techniques for Engineering Design with Applications. New York: McGraw-Hill Book Company. Van Dyke, M. D. (1952) . A Study of Second-Order Supersonic Flow Theory. NACA Report 1081. Yates, E. C., Jr., and Bennett, R. M. (1963). Use of Aerodynamic Parameters from Nonlinear Theory in Modified-Strip-Analysis Flutter Calculations for Finite-Span Wings at Supersonic Speeds, NASA TND-1824. Zartarian, G. (1956) . Theoretical Studies on the Prediction of Unsteady Airloads on Elastic Wings; Part II - Rules for Application of Oscillatory Supersonic Aerodynamic Influence Coefficients. Wright Air Development Center, WADC Technical Report 56-97, Part II. Zartarian, G., and Hsu, P. T. (1955) . Theoretical Studies on the Prediction of Unsteady Airloads on Elastic Wings; Part I - Investigations on the Use of Oscillatory Supersonic Aerodynamic Influence Coefficients. Wright Air Development Center, WADC Technical Report 56-97, Part I.

A.2

Bibliography of Aeroelasticity and Unsteady Aerodynamics

The available literature on the structural, dynamic, and aerodynamic aspects of aeroelasticity is extensive, and it is assumed that the user has some familiarity with these topics. The literature dedicated exclusively to aeroelasticity is also extensive. However, the following list of textbook references on aeroelasticity and unsteady aerodynamics can be recommended to the

A-4

NX Nastran Aeroelastic Analysis User’s Guide

References and Bibliography

user for guidance on the fundamentals of aerodynamic/structural interaction. Two optimization references are also cited to provide an introduction to the wide range of literature in this field. All references are listed in chronological order. 1. Freberg, C. R., and Kemler, E. N. (1944). Aircraft Vibration and Flutter . New York: John Wiley & Sons. 2. Perkins, C. D., and Hage, R. E. (1949). Airplane Performance, Stability and Control . New York: John Wiley & Sons. 3. Scanlan, R. H., and Rosenbaum, R. (1951). Introduction to the Study of Aircraft Vibration and Flutter . New York: The Macmillan Company. 4. Bisplinghoff, R. L., Ashley, H., and Halfman, R. L. (1955). Aeroelasticity. Reading: Addison-Wesley Publishing Company. 5. Fung, Y. C. (1955). An Introduction to the Theory of Aeroelasticity . New York: John Wiley & Sons. 6. Den Hartog, J. P. (1956). Mechanical Vibrations . New York: McGraw-Hill Book Company, 4th ed. 7. Garrick, I. E. (1957) “Nonsteady Wing Characteristics,” Sect. F., Vol. VII, of High Speed Aerodynamics and Jet Propulsion . Princeton N.J.: Princeton University Press. 8. Abramson, H. N. (1958). An Introduction to the Dynamics of Airplanes . New York: The Ronald Press Company. 9. Miles, J. W. (1958). The Potential Theory of Unsteady Supersonic Flow . Cambridge: The University Press. 10. Many authors, Manual on Aeroelasticity , Vols. I-VII, Advisory Group for Aerospace Research and Development (AGARD), beginning in 1959 with continual updating. 11. Landahl, M. T. (1961). Unsteady Transonic Flow . Oxford: Pergamon Press. 12. Bisplinghoff, R. L., and Ashley, H. (1962). Principles of Aeroelasticity. New York: John Wiley & Sons. 13. Ashley, H., and Landahl, M. (1965). Aerodynamics of Wings and Bodies . Reading: Addison-Wesley Publishing Company. 14. Petre, A. (1966). Teoria Aeroelasticitiatii-Statica . Bucharest: Editura Academiei Republicii Socialiste Romania. 15. Garrick, I. E. (1969). Aerodynamic Flutter . New York: AIAA Selected Reprint Series, Vol. V, 1969. 16. Ashley, H., and Rodden, W. P. (1972). “Wing-Body Interaction,” pp. 431-471, Annual Review of Fluid Mechanics . Palo Alto: Annual Reviews, Inc., Vol. 4. 17. Etkin, B. (1972). Dynamics of Atmospheric Flight . New York: John Wiley & Sons. 18. Petre, A. (1973). Teoria Aeroelasticitiatii-Fenomene Dinamice Periodice . Bucharest: Editura Academici Republicii Socialiste Romania.

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A-5

Appendix A

References and Bibliography

19. Försching, H. W. (1974). Grundlagen der Aeroelastik . Berlin: Springer-Verlag. 20. Blevins, R. D. (1977). Flow-induced Vibration . New York: Van Nostrand Reinhold Company. 21. Vanderplaats, G. N. (1984). Numerical Optimization Techniques for Engineering Design with Applications . New York: McGraw-Hill Book Company. 22. Haftka, R. T., and Kamat, M. P. (1985). Elements of Structural Optimization . The Hague: Martinus Nijhoff Publishers. 23. Hoblit, F. M. (1988). Gust Loads on Aircraft: Concepts and Applications . Washington, D.C.: American Institute of Aeronautics and Astronautics. 24. Dowell, E. H., Curtiss, H. C., Jr., Scanlan, R. H., and Sisto, F. (1990). A Modern Course in Aeroelasticity . Alphen aan der Rijn, The Netherlands: Sijthoff & Noordhoff, 2nd ed.

A.3

Bibliography of MSC.Nastran Applications

This bibliography is a summary of MSC.Nastran-related papers dealing with solutions to aeroelastic and aerodynamic problems. They are excerpted from the MSC.Nastran Bibliography (2nd ed.) and are gathered from various MSC Users’ Conferences, other NASTRAN User’s Colloquia, and a search of the general engineering literature. AEROELASTICITY / AERODYNAMICS Armand, Sasan C.; Liao, Mei-Hwa; Morris, Ronald W. A Transient Plasticity Study and Low Cycle Fatigue Analysis of the Space Station Freedom Photovoltaic Solar Array Blanket , The MSC 1990 World Users Conf. Proc., Vol. I, Paper No. 19, March, 1990. Bhatia, Kumar G.; Nagaraja, K. S. Transonic and Density Effect Investigations , Flutter Parametric Studies of Cantilevered Twin-Engine-Transport Type Wing with and without Winglet, Vol. II, September, 1984, (NASA CR-172410). Boroughs, Robert R. Development of the LEARJET 28/29 Wing Using NASTRAN Analysis, Eighth NASTRAN Users’ Colloq., pp. 11-32, October, 1979, (NASA CP-2131). Briggs, Hugh C.; Chrisinger, Lance E. A New Capability for Elastic Aircraft Airloads via NASTRAN, Tenth NASTRAN Users’ Colloq., pp. 165-186, May, 1982, (NASA CP-2249). Cifuentes, Arturo; Bock, Timothy L.; Coppolino, Robert N. A Finite Element Technique for Tape-Head Interaction Problems in High Speed Recording: The Steady-State Case, The 1989 MSC World Users Conf. Proc., Vol. I, Paper No. 24, March, 1989. Clark, Wm. B. Gulfstream Experience Using NASTRAN Solution 76 for Lateral Gust Loads Analysis of the G-IV Airplane, Proc. of the MSC/NASTRAN Aeroelasticity Users Conf., Paper No. 1, October, 1989. Climent, Hector; Johnson, Erwin H. Aeroelastic Optimization Using MSC/NASTRAN , Int. Forum on Aeroelasticity and Structural Dynamics 1993, May, 1993. Crimaldi, J. P.; Britt, R. T.; Rodden, W. P. Response of the USAF/Northrop B-2 Aircraft to Nonuniform Spanwise Atmospheric Turbulence, Proc. of the 32nd AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Paper No. 91-1048, April, 1991. Crittenden, J. B.; Weisshaar, T. A.; Johnson, E. H.; Rutkowski, M. J. Aeroelastic Stability Characteristics of an Oblique-Wing Aircraft, J. of Aircraft, Vol. 15, No. 7, pp. 429-434, July, 1978.

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NX Nastran Aeroelastic Analysis User’s Guide

References and Bibliography

Deerhake, Allen. A Critique of the MSC/NASTRAN Flutter Analysis Capability, Proc. of the MSC/NASTRAN Users’ Conf., March, 1979. Desy, Philippe; Perron, Claude. Loads Balancing Technique Using MSC/NASTRAN Results, The MSC 1991 World Users’ Conf. Proc., Vol. I, Paper No. 16, March, 1991. Doggett, Robert V.; Harder, Robert L. Subsonic Flutter Analysis Addition to NASTRAN, NASTRAN: Users’ Exper., pp. 507-530, September, 1973, (NASA TM X-2893). Doggett, Robert V.; Cunningham, Herbert J. Jr. Some Applications of the NASTRAN Level 16 Subsonic Flutter Analysis Capability, NASTRAN: Users’ Exper., pp. 495-512, October, 1976, (NASA TM X-3428). Eversman, W.; Tewari, A. A Reduced Cost Rational-Function Approximation for Unsteady Aerodynamics, Proc. of the MSC/NASTRAN Aeroelasticity Users Conf., Paper No. 11, October, 1989. Geyer, A.; Schweiger, W. Aeroelastic and Stress Analysis of the CHIWEC Chinese Wind Energy Converter Using MSC/NASTRAN, Proc. of the MSC/NASTRAN Eur. Users’ Conf., April, 1985. Harder, R. L.; MacNeal, R. H.; Rodden, W. P. A Design Study for the Incorporation of Aeroelastic Capability into NASTRAN, National Aeronautics and Space Administration, May, 1971, (NASA CR-111918). Harder, R. L.; MacNeal, R. H.; Rodden, W. P. A Study of a Finite Element Method to Calculate Oscillatory Supersonic Lift Distributions Using the Acceleration Potential Kernel, National Aeronautics and Space Administration, August, 1971, (NASA CR-111998). Harder, Robert L.; MacNeal, Richard H.; Doggett, Robert V. Jr. A Design Study for the Incorporation of Aeroelastic Capability, NASTRAN: Users’ Exper., pp. 779-796, September, 1971, (NASA TM X-2378). Heinze, P.; Schierenbeck, D.; Niemann, L. Structural Optimization in View of Aeroelastic Constraints Based on MSC/NASTRAN FE-Calculation, Proc. of the Sixteenth MSC Eur. Users’ Conf., Paper No. 15, September, 1989. Hsu, Lu-Jiin; Chiou, Hua-Tzong; Chen, Kuo-Her. A Study of the Deformation Behavior of Wing for Composite Material, The 2nd Annual MSC Taiwan Users Conf., Paper No. 4, October, 1990. Hwang, Jyh-Yeon. A Study of the Structural Dynamic Characteristics Under the Aeroelastic Interaction , The 4th MSC Taiwan Users’ Conf., Paper No. 10, November, 1992, in Chinese. Jew, Howard. Delta Wing Flutter Based on Doublet Lattice Method in NASTRAN, NASTRAN: Users’ Exper., pp. 603-614, September, 1975, (NASA TM X-3278). Johnson, Erwin H. Further Development of the Aeroelasticity Capability in MSC/NASTRAN, Proc. of the MSC/NASTRAN Aeroelasticity Users Conf., Paper No. 8, October, 1989. Johnson, E. H.; Rodden, W. P. Ongoing Development of the MSC/NASTRAN Aeroelastic Capability, Proc. of the 17th MSC Eur. Users’ Conf., Paper No. 10, September, 1990. Johnson, Erwin; Nagendra, Gopal; Reymond, Michael. Addition of Aeroelasticity to the MSC/NASTRAN Multidisciplinary Analysis and Design Capability, The MSC 1991 World Users’ Conf. Proc., Vol. I, Paper No. 7, March, 1991. Karpel, Mordechay; Presente, Eyal. Structural Dynamic Loads in Response to Impulsive Excitation , Int. Forum on Aeroelasticity and Structural Dynamics 1993, May, 1993. Kooj, B.; Wijker, J. Air Effects on the Dynamic Behaviour of Stowed Solar Array Wings, Proc. of the Fifteenth MSC/NASTRAN Eur. Users’ Conf., October, 1988.

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A-7

Appendix A

References and Bibliography

Lecce, Leonardo; Marulo, Francesco; D’Amato, Ciro Angelo; Verde, Giuseppe. Aeroelastic Behaviour on an All-Movable Horizontal Tail of a General Aviation Twin-Engine Aircraft , Int. Forum on Aeroelasticity and Structural Dynamics 1993, May, 1993. Lee, In; Miura, Hirokazu; Chargin, Mladen; Green, John. Static Aeroelastic Analysis for Generic Configuration Wing, Proc. of the MSC/NASTRAN Aeroelasticity Users Conf., Paper No. 2, October, 1989. Lee, Kuan S.; Miller, R. P.; Porada, T. W. An Application of MSC/NASTRAN Transfer Functions for Launch Vehicle Air-Loads Analysis, The MSC 1991 World Users’ Conf. Proc., Vol. I, Paper No. 15, March, 1991. Lenselink, H. Aerodynamic Loading on a Thrust-Reverser, The MSC 1991 World Users’ Conf. Proc., Vol. I, Paper No. 17, March, 1991. Lewis, A. P. A NASTRAN DMAP Procedure for Aeroelastic Design Sensitivity Analysis, Proc. of the 18th MSC Eur. Users’ Conf., Paper No. 17, June, 1991. Lin, Hung-Hsi; Jhou, Jitai; Stearman, Ronald. Influence of Joint Fixity on the Structural Static and Dynamic Response of a Joined Wing Aircraft, Proc. of the MSC/NASTRAN Aeroelasticity Users Conf., Paper No. 9, October, 1989. Liu, D. D. ZONA Codes for Supersonic Aeroelastic Application, Proc. of the MSC/NASTRAN Aeroelasticity Users Conf., Paper No. 10, October, 1989. MacNeal, R. H.; Rodden, W. P. Requirements for the Addition of Aeroelastic Capability to NASTRAN, The MacNeal-Schwendler Corporation, Report No. MSR-21, March, 1970. Masse, Barnard; Pastorel, Henri. Stress Calculation for the Sandia 34-Meter Wind Turbine Using the Local Circulation Method and Turbulent Wind, The MSC 1991 World Users’ Conf. Proc., Vol. II, Paper No. 53, March, 1991. McNamee, Martin J.; Zavareh, Parviz. Nonlinear Transient Analysis of a Shock Isolated Mechanical Fuse, The MSC 1990 World Users Conf. Proc., Vol. I, Paper No. 21, March, 1990. Mei, Chuh; Rogers, James L. Jr. Application of NASTRAN to Large Deflection Supersonic Flutter of Panels, NASTRAN: Users’ Exper., pp. 67-98, October, 1976, (NASA TM X-3428). Nakamura, Hitoshi; Ogaki, Ikuko. Flutter Simulation for Bridges, The MSC 1990 World Users Conf. Proc., Vol. I, Paper No. 20, March, 1990. Parthasarathy, Alwar. Force-Sum Method for Dynamic Stresses in MSC/NASTRAN Aeroelastic Analysis, The MSC 1991 World Users’ Conf. Proc., Vol. I, Paper No. 8, March, 1991. Rodden, W. P.; Harder, R. L.; Bellinger, E. D. Aeroelastic Addition to NASTRAN, National Aeronautics and Space Administration, March, 1979, (NASA CR-3094). Rodden, W. P.; Bellinger, E. D. Aerodynamic Lag Functions, Divergence, and the British Flutter Method, J. of Aircraft, Vol. 19, No. 7, pp. 596-598, 1982. Rodden, W. P.; Bellinger, E. D. Unrestrained Aeroelastic Divergence in a Dynamic Stability Analysis, J. of Aircraft, Vol. 19, No. 9, pp. 796-797, 1982. Rodden, W. P.; Bellinger, E. D.; Giesing, J. P. Errata and Addenda to ‘Application of Oscillatory Aerodynamic Theory to Estimation of Dynamic Stability Derivatives,’ J. of Aircraft, Vol. 21, No. 1, pp. 93-94, 1984. Rodden, W. P.; Wilson, C. T.; Herting, D. N.; Bellinger, E. D.; MacNeal, R. H. Static Aeroelastic Addition to MSC/NASTRAN, MSC/NASTRAN Users’ Conf. Proc., Paper No. 15, March, 1984. Rodden, W. P. Secondary Considerations of Static Aeroelastic Effects on High-Performance Aircraft, Static Aeroelasticity in Combat Aircraft, Paper No. 3, January, 1986, (AGARD Report 725).

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NX Nastran Aeroelastic Analysis User’s Guide

References and Bibliography

Rodden, William P., (ed.) Handbook for Aeroelastic Analysis , The MacNeal-Schwendler Corporation, Report No. MSR-57, November, 1987. Rodden, William P. The New MSC/NASTRAN Handbook for Aeroelastic Analysis, The MSC 1988 World Users Conf. Proc., Vol. II, Paper No. 63, March, 1988. Turner, Daniel, Dr.; Astle, Donald C. A Study of the Effect of Control Surface Aerodynamics on Flutter Analyses, Proc. of the Conf. on Finite Element Methods and Technology, Paper No. 16, March, 1981. Tzong, George T.J.; Sikes, Gregory D.; Dodd, Alan J. Large Order Modal Analysis Module in the Aeroelastic Design Optimization Program (ADOP) The MSC 1991 World Users’ Conf. Proc., Vol. II, Paper No. 36, March, 1991. Visagan, G. Senthil; Elayathu, M. N. G. Flutter Analysis of Launch Vehicle Fin , The First India MSC Users’ Conf. Proc., December, 1992. Vollan, A. Use of MSC/NASTRAN for the Aeroelastic Analysis of the Pilatus PlC-9 Turbo Trainer, Proc. of the MSC/NASTRAN Eur. Users’ Conf., May, 1986. Vollan, Arne; Alati, Loreto; Ronti, Gino. Unsteady Pressure Distribution of a Wing-Body Configuration: MSC/NASTRAN Results Compared with Wind Tunnel Tests, The MSC 1990 World Users Conf. Proc., Vol. I, Paper No. 22, March, 1990. Weisshaar, Terrence A.; Sadlowski, Michael. Panel Flutter Suppression with Active Micro-Actuators , Int. Forum on Aeroelasticity and Structural Dynamics 1993, May, 1993. Winther, B. A.; Hagemeyer, D. A.; Britt, R. T.; Rodden, W. P. Aeroelastic Effects on the B-2 Maneuver Response , AIAA Atmospheric Flight Mechanics Meeting, August, 1993, (AIAA-93-3664). Zwillenberg, Peter A. MSC/NASTRAN Doublet-Lattice Method Qualification and Correlation, The MSC 1991 World Users’ Conf. Proc., Vol. I, Paper No. 9, March, 1991.

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Appendix

B Commonly Used Commands for Aeroelasticity

•

Executive Control Commands

•

Case Control Commands

•

Aerodynamic Trim Variable Selection

•

Divergence Solution Selection

•

Structural Damping and Transfer Function Selection

•

Aeroelastic Flutter Solution Selection

•

Aeroelastic Dynamic Load Selection

•

Output Control

•

Bulk Data Entries

•

Bulk Data Entry Summary

•

Parameters

NX Nastran Aeroelastic Analysis User’s Guide

B-1

Appendix B

B.1

Commonly Used Commands for Aeroelasticity

Executive Control Commands

The aeroelastic analysis and design solution sequences are: SOLs 21 and 144 SOLs 75 and 145

Static Aeroelastic Response Superelement Aerodynamic Flutter

SOLs 76 and 146

Superelement Aeroelastic Response

SOL 200

Optimization

The restart capability of the aeroelastic DMAP sequences is controlled by parameters and superelement processing requests analogous to the other DMAP sequences.

B.2

Case Control Commands

The Case Control Section is described in the Case Control Commands in the NX Nastran Quick Reference Guide. The following Case Control data selection commands are available for aeroelastic analyses and design and various outputs:

B.3

Aerodynamic Trim Variable Selection Selects a TRIM Bulk Data entry in static aeroelastic response

Trim

B.4

Divergence Solution Selection

DIVERG

B.5

Structural Damping and Transfer Function Selection

SDAMPING TFL

B.6

Selects the number of eigenvalues and Mach numbers for the aeroelastic divergence analysis

Selects table that defines damping as a function of frequency Selects the transfer function set that represents a servomechanism

Aeroelastic Flutter Solution Selection

FMETHOD CMETHOD

Selects the method to be used in aeroelastic flutter analysis Selects complex eigenvalue method for flutter analysis

METHOD

Selects real eigenvalue method for vibration analysis

B-2

NX Nastran Aeroelastic Analysis User’s Guide

Commonly Used Commands for Aeroelasticity

B.7

Aeroelastic Dynamic Load Selection

DLOAD FREQUENCY GUST RANDOM TSTEP

B.8

Output Control

AEROF APRESSURE SET

B.9

Selects the dynamic load to be applied in a transient or frequency response problem Selects the set of frequencies to be solved in frequency response problems Selects the gust load in aeroelastic response analysis Selects the RANDPS and RANDTi entries to be used in random analysis Selects integration and output time steps for transient problems

Requests the aerodynamic loads on the aerodynamic control points Requests the aerodynamic pressures in static aeroelastic response Lists identification numbers for output requests or lists frequencies for which output will be printed in frequency response analysis

Bulk Data Entries

The Bulk Data entries are described in Bulk Data Entries in the NX Nastran Quick Reference Guide. The format of the aeroelastic Bulk Data entries is the standard Bulk Data input format described at the beginning of Bulk Data Entries in the NX Nastran Quick Reference Guide. Refer to the NX Nastran Quick Reference Guide for current features for each new application.

B.10

Bulk Data Entry Summary AERO AEROS GUST

AEFACT CAERO1 CAERO2 CAERO3 CAERO4 CAERO5 PAERO1 PAERO2 PAERO3 PAERO4 PAERO5

Aerodynamic Parameters Aerodynamic physical parameters for flutter or dynamic aeroelastic response analysis and design Static aeroelasticity physical properties Defines aerodynamic gust load Aerodynamic Geometry and Element Specifies lists of real numbers Connection definition for subsonic or ZONA51 aerodynamic panel Connection definition for subsonic slender body Connection definition for Mach Box theory Connection definition for aerodynamic macro-strip Connection definition for Piston Theory Defines associated bodies for subsonic panel Property definition for subsonic slender body Property definition for Mach Box theory Property definition for strip elements Property definition for Piston Theory strips

NX Nastran Aeroelastic Analysis User’s Guide

B-3

Appendix B

MKAER01 MKAER02 SET1 SET2 SPLINE1 SPLINE2 SPLINE3 TABDMP1 DMIG TF AELINK AELIST AESTAT AESURF DIVERG SUPORT TRIM EIGC EIGR EIGRL FLFACT FLUTTER DAREA DELAY DLOAD DPHASE FREQi TLOADi TSTEP RANDPS RLOADi TABRNDG

B-4

Commonly Used Commands for Aeroelasticity

Aerodynamic Matrix Generation Table of Mach numbers and reduced frequencies Table of Mach numbers and reduced frequencies Structure/Aerodynamic Interconnection Defines a set of structural grid points Defines a set of structural grid points in terms of aerodynamic macro-elements Defines a surface spline for interpolating out-of-plane motion Defines a beam spline for interpolating panels and bodies Defines a constrain equation Structure Damping Defines structural damping as a tabular function of frequency Servomechanisms Defines structure-related direct input matrices Defines dynamic transfer function Static Aerolastic Analysis Defines relationships among AESTAT and AESURF entries Defines list of aerodynamic elements Specifies rigid body motions to be used as trim variables Specifies aerodynamic control surfaces Specifies number of divergence eigenvalues and Mach numbers Defines coordinates for determining reactions Specifies contraints for aeroelastic trim variables Vibration and Flutter Analyses Defines data needed to perform complex eigenvalue analysis Defines data needed to perform real eigenvalue analysis Defines data needed to perform real eigenvalue analysis with the Lanczos Method Specifies aerodynamic data for flutter analysis Specifies data to perform flutter analysis Dynamic Aeroelastic Response Dynamic load scale factor definition Dynamic load function time delay definition Linear combination of dynamic load sets Dynamic load function phase lead definition Defines a set of frequencies for problem solutions Transient Aeroelastic Response Analysis Time dependent load definition Specifies time step intervals for solution and output Frequency and Random Aeroelastic Response Analysis Defines load and set power spectral density factors Frequency dependent load definition Defines the power spectral density of a gust

NX Nastran Aeroelastic Analysis User’s Guide

Commonly Used Commands for Aeroelasticity

TABRND1 DRESP1 DSCREEN

B.11

Defines the power spectral density as a tabular function of frequency Design Sensitivity and Optimization Defines a set of structural responses that is used in the design either as a constraint or as an objective Defines constraint screening data for constraint deletion

Parameters

PARAM Bulk Data entries are used in the DMAP sequences for input of miscellaneous data and to request special features. The most important parameters used in aeroelastic analysis and design are described in alphabetical order. The complete description of all parameters is contained in Parameters in the NX Nastran Quick Reference Guide , and the used should refer to that section for current features Default = 0.0

G

GUSTAERO

HFREQ

KDAMP

LFREQ

G specifies the uniform structural damping coefficient in the formulation of dynamics problems. To obtain the value for the parameter G, multiply the critical damping ratio, C/C0 ,by 2.0. PARAM,G is not recommended for use in hydroelastic or heat problems. If PARAM,G is used in transient analysis, PARAMW3 must be greater than zero or PARAM,G will be ignored (see Section 9.4.10 of the MSC.Nastran Reference Manual). Default = 1 If gust loads are not to be computed, for example on restart, set GUSTAERO to -1. The default is recommended if no gust loads are to be computed after the flutter analysis. Default = 1.+30 The parameters LFREQ and HFREQ specify the frequency range in cycles per unit time (LFREQ is the lower limit and HFREQ is the upper limit) of the modes to be used in the modal formulations. Note that the default for HFREQ will usually include all vectors computed. A related parameter is LMODES. Default = 1 If KDAMP is set to -1, viscous modal damping is entered into the complex stiffness matrix as structural damping. See Section 9.4.10 of the MSC.Nastran Reference Manual. Default = 0.0 See HFREQ.

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Appendix B

Commonly Used Commands for Aeroelasticity

Default = 0 LMODES is the number of lowest modes to use in a modal formulation. If LMODES = 0, the retained modes are determined by the parameters LFREQ and HFREQ. In coupled fluid-structure anaylysis, LMODES is applied only to the structural portion of the model. Default = 0.0

LMODES

Mach number. If more than one MACH number was used to compute aerodynamic matrices, the one closest to MACH will be used in dynamic aeroelastic response analysis. The default causes the matrices computed at the lowest MACH number to be used. Default = -1

MACH

OPGTKG>-1prints the matrix for the interpolation between the structural and aerodynamic degrees of freedom. Default = -1

OPGTKG

This parameter is valid in the flutter (SOLs75 and 145) and aerolastic (SOLs76 and 146) solution sequences. OPPHIPA >-1 and a DISPLACEMENT command in the Case Control Section will output the real vibration modes at all degrees of freedom, including the aerodynamic degrees of freedom. Default = 0

OPPHIPA

In SOL200, especially during the checkout of the analysis model and the design optimization input data (design model), it may be desirable to exit the solution sequence at certain points before proceeding with full optimization. OPTEXIT may be set to values of 1 through 6 and -4. The description of OPTEXIT values follow.

OPTEXIT

OPTEXIT Value 0 1 2 3 4

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Description Do not exit. Proceed with optimization. Exit after the initialization of the analysis and design model but before finite element analysis begins. Exit after finite element analysis and initial design response and shape basis vector processing. Exit after design constrain evaluation and screening. Exit after design sensitivity analysis and print the matrix of design sensitivity coefficients (DSCM2).

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Commonly Used Commands for Aeroelasticity

-4 5 6

Exit after design sensitivity analysis and write the data blocks related to sensitivity coefficients (DSCM2 and DSCMCOL) to an external file using the OUTPUT2 and OUTPUT4 modules. See related parameters ITAPE, IUNIT, and OMAXR. Exit after the first approximation optimization of the design model. Exist after the first update of the analysis model based on the first approximate optimization of the design model. Default = YES

PRINT

Q

PARAM,PRINT,NO suppress the automatic printing of the flutter summary in flutter analysis. Default = 0.0 Q specifies the dynamic pressure. Q must be specified in aeroelastic response analysis (SOLutions 76 and 146) and the default value will cause a User Fatal Message. Default = 0 For Solutions 145, 146, and 200, SKPAMP = -1 suppresses all unsteady aerodynamic calculations. The automatic restart of structural solution sequences performs a similar function without this parameter. Specifying it ensures the suppression of the calculations, regardless of the determination of the automatic restart.

SKPAMP

For Solutions 75 and 76, SKPAMP<0 suppresses the calculation of the generalized aerodynamic matrices. The use should specify SKPAMP = -1 whenever a restart is to use the real vibration modes and aerodynamic matrices from a previous run. Related parameters include DLOAD, NEWDYN, NODATA, PLOT, RESDUAL, and SDATA. These entries must be resubmitted in the Bulk Data Section for restart runs. Default = 1.0

VREF

In modal flutter analysis, the velocities are divided by VREF to convert units or to compute flutter indices.

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